# Go Math Grade 7 Answer Key Chapter 7 Writing and Solving One-Step Inequalities

## Go Math Grade 7 Answer Key Chapter 7 Writing and Solving One-Step Inequalities

It is important to gain knowledge along with the marks. Go Math Answer Key team’s main aim is to provide quality education for students of all grades. You can learn the basics of Writing and Solving One-Step Inequalities in Go Math Grade 7 Chapter 7 Answer key. Check out the topics before you start practicing. Only practice will help you to score the best marks in the exams. Refer to our Go Math Grade 7 Answer Key Chapter 7 Writing and Solving One-Step Inequalities while doing your homework and also during exam preparation. You can know how to draw the number line with the help of Go Math 7th Grade Chapter 7 Writing and Solving One-Step Inequalities.

Chapter 7 – Lesson 1:

Chapter 7 – Lesson: 2

Chapter 7 – Lesson: 3

Chapter 7 – Writing and Solving One-Step Inequalities Lesson: 4

Chapter 7 – Lesson: 5

### Guided Practice – Page No. 208

Write the resulting inequality.

Question 1.
−5 ≤ −2; Add 7 to both sides
Type below:
___________

Explanation:
Add 7 to both sides of the inequality.
-5 + 7 ≤ -2 + 7
2 ≤ 5

Question 2.
−6 < −3; Divide both sides by -3
Type below:
___________

Explanation:
Divide both sides by -3. switch the inequality sign since you are dividing by a negative number.
-6/-3 > -3/-3
2 > 1  Solve each inequality. Graph and check the solution.

Question 5.
n−5 ≥ −2
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
n – 5 ≥ -2
n – 5 + 5 ≥ -2 + 5
n ≥ 3
The number opposite the variable is 3, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would look like the one below:

Question 6.
3 + x < 7
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
3 + x < 7
3 – 3 + x < 7 – 3
x < 4
The number opposite the variable is 4, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would look like the one below:

Question 7.
−7y ≤ 14
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
−7y ≤ 14
-7y/-7 ≤ 14/-7
y ≥ -2
The number opposite the variable is -2, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would look like the one below:

Question 8.
$$\frac{b}{5}$$ > −1
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
Solve the inequality first:
$$\frac{b}{5}$$ > −1
Multiply 5 on both sides.
(5)$$\frac{b}{5}$$ > −1(5)
b > -5
The number opposite the variable is -5, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would look like the one below:

Question 9.
For a scientific experiment, a physicist must make sure that the temperature of a metal at 0 °C gets no colder than -80 °C. The physicist changes the metal’s temperature at a steady rate of -4 °C per hour. For how long can the physicist change the temperature?
a. Let t represent the temperature in degrees Celsius. Write an inequality. Use the fact that the rate of change in temperature times the number of hours equals the final temperature.
Type below:
___________

We need to use the fact that the final temperature is equal to the rate of change in temperature times the number of hours.
We are given that the rate of change is -4°C per hour so the final temperature is -4 times the number of hours.
Let t represent the number of hours. The final temperature is then -4t degrees Celsius after t hours.
If the temperature must be no colder than -80°C, then the final temperature must be greater than or equal to -80.
The inequality is then -4t ≥ -80.

Question 9.
b. Solve the inequality in part a. How long can the physicist change the temperature of the metal?
Type below:
___________

To solve the inequality for t, we need to divide both sides by -4. Remember to switch the inequality symbol since you are dividing by a negative number.
Dividing both sides by -4 then gives:
-4t/-4 ≤ -80/-4
t ≤ 20
The number of hours that the physicist can change the temperature of the metal is then at most 20 hours.

Question 9.
c. The physicist has to repeat the experiment if the metal gets cooler than -80 °C. How many hours would the physicist have to cool the metal for this to happen?
Type below:
___________

From part (b), we know that the physicist can change the temperature for at most 20 hours to keep the temperature no colder than -80°C. This means the temperature will reach a temperature cooler than -80°C if he cools the metal for more than 20 hours.

Essential Question Check-In

Question 10.
Suppose you are solving an inequality. Under what circumstances do you reverse the inequality symbol?
Type below:
___________

Answer: You must reverse the inequality sign any time you multiply or divide both sides of the inequality by a negative number.

### Page No. 209

In 11–16, solve each inequality. Graph and check the solution.

Question 11.
x − 35 > 15
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
x − 35 > 15
x – 35 + 35 > 15 + 35
x > 50
The number opposite the variable is 50, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would like the one below: Question 13.
−$$\frac{q}{7}$$ ≥ −1
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−$$\frac{q}{7}$$ ≥ −1
Multiply both sides by -7
(-7)−$$\frac{q}{7}$$ ≥ −1(-7)
q ≤ 7
The number opposite the variable is 7, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would like the one below:

Question 14.
−12x < 60
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
−12x < 60
Divide both sides by -12
-12x/12 < 60/-12
x > -5
The number opposite the variable is -5, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would like the one below:

Question 15.
5 > z − 3
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
5 > z − 3
5 + 3 > z – 3 + 3
8 > z
z < 8
The number opposite the variable is 8, we look for this in the number line. Since the inequality is <, we use a hollow dot and shade the line going to the left. Its graph would like the one below:

Question 16.
0.5 ≤ $$\frac{y}{8}$$
Type below:
___________

To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution.
From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >.
First, solve the inequality:
0.5 ≤ $$\frac{y}{8}$$
Multiply both sides by 8
(8)0.5 ≤ $$\frac{y}{8}$$(8)
4 ≤ y
y ≥ 4
The number opposite the variable is 4, we look for this in the number line. Since the inequality is ≥, we use a closed dot and shade the line going to the right. Its graph would like the one below:

Question 17.
The vet says that Lena’s puppy will grow to be at most 28 inches tall. Lena’s puppy is currently 1 foot tall. How many more inches will the puppy grow?
Type below:
___________

Answer: not more than 16 inches

Explanation:
Let x be the additional inches the puppy can grow remember 1 foot is 12 inches so the height of the puppy is 12 + x
12 + x ≤ 28
x ≤ 16

Question 18.
In a litter of 7 kittens, each kitten weighs less than 3.5 ounces. Find all the possible values of the combined weights of the kittens.
Type below:
___________

All of the kittens must weigh more than 0 ounces so the smallest combined weight is more than 0 ounces. Since there are 7 kittens, each kitten weights less than 3.5 ounces, and 7 × 3.5 = 24.5, then the combined weights of the kittens must be less than 24.5 ounces.
This gives the inequality 0 < w < 24.5
where w is the combined weight of the kittens in ounces.

Question 19.
Geometry
The sides of the hexagon shown are equal in length. The perimeter of the hexagon is at most 42 inches. Find the possible side lengths of the hexagon. Type below:
___________

Answer: 0 < s ≤ 7

Explanation:
Let s be the side lengths of the hexagon since its sides are all equal in length.
The side lengths of the hexagon must be greater than 0 since lengths can’t be negative or 0 so s > 0.
The perimeter of the figure is the sum of its side lengths so the perimeter of the hexagon must be 6s since it has 6 sides that are all s inches long.
The perimeter is at most 42 inches so 6s ≤ 42.
Dividing both sides by 6 then gives s ≤ 7.
Combining the inequalities s > 0 and s ≤ 7 then gives possible side lengths of 0 < s ≤ 7.  Question 22.
Draw Conclusions
A submarine descends from sea level to the entrance of an underwater cave. The elevation of the entrance is -120 feet. The rate of change in the submarine’s elevation is no greater than -12 feet per second. Can the submarine reach the entrance to the cave in less than 10 seconds? Explain.
Type below:
___________

No. Since the rate of descent is less than -12 feet per second and the submarine is descending for less than 10 seconds, the submarine elevation will still be greater than -120. The submarine would have to descend at a rate greater than -12 feet per second to reach the entrance in less than 10 seconds or descend for more than 10 seconds at a rate less than -12 feet per second to reach the entrance.

### Page No. 210

The sign shows some prices at a produce stand. Question 23.
Selena has $10. What is the greatest amount of spinach she can buy? Type below: ___________ Answer: 3 $$\frac{1}{3}$$ pounds Explanation: Let x be the number of pounds of spinach. divide both sides by 3 to solve for x. 3x ≤ 10 x ≤ $$\frac{10}{3}$$ x ≤ 3 $$\frac{1}{3}$$ pounds Question 25. Florence wants to spend no more than$3 on onions. Will she be able to buy 2.5 pounds of onions? Explain.
Type below:
___________

Since each pound of onions costs $1.25, then 2.5 pounds of onions cost$1.25 × 2.5 ≈ 3.13.
Since $3.13 is greater than$3, she will not have enough money if she wants to spend no more than $3. H.O.T. Focus on Higher Order Thinking Question 26. Counterexamples John says that if one side of an inequality is 0, you don’t have to reverse the inequality symbol when you multiply or divide both sides by a negative number. Find an inequality that you can use to disprove John’s statement. Explain your thinking. Type below: ___________ Answer: A possible counterexample is -2x ≤ 0. Solving this correctly gives x ≥ 0 which means the inequality is true for all non-negative values. If you don’t switch the inequality sign you would get x ≤ 0 which means the inequality would be true for all non-positive numbers x = -3 is a possible value for x ≤ 0 but -2x = -2(-3) = 6 which is not less than or equal to 0. Question 28. Persevere in Problem-Solving The base of a rectangular prism has a length of 13 inches and a width of $$\frac{1}{2}$$ inch. The volume of the prism is less than 65 cubic inches. Find all possible heights of the prism. Show your work. Type below: ___________ Answer: 0 < h < 10 Explanation: Let h be the height of the prism. It is given that the prism has a length of 13 inches and a width of 1/2 inch. Using the formula v = lbh 13(1/2) h< 65 Multiply 2/13 on both sides 2/13 (13/2)h< 2/13 × 65 h < 2 × 5 h < 10 Since the height must be a positive number, then h > 0. Combining h > 0 and h < 10 then gives the final answer of 0 < h < 10. ### Guided Practice – Page No. 214 Draw algebra tiles to model each two-step inequality. Question 1. 4x − 5 < 7 Type below: ___________ Answer: On the left side, draw 4 positive rectangles to model 4x and 5 negative squares to represent -5. On the right side, draw 7 positive squares to represent 7. then draw < in the middle. Question 2. −3x + 6 > 9 Type below: ___________ Answer: On the left side, draw 3 negative rectangles to model -3x and 6 positive squares to represent 6. On the right side, draw 9 positive squares to represent 9. then draw > in the middle. Question 3. The booster club needs to raise at least$7,000 for new football uniforms. So far, they have raised $1,250. Write an inequality to find the average amounts each of the 92 members can raise to meet the club’s objective. Type below: ___________ Answer: 1250 + 92a ≥ 7000 Explanation: The amount to be raised is$7000. The amount already raised is $1250. The number of members is 92. The inequality is then of the form: amount already raised + number of members × amount each member raises ≥ target amount. The inequality is then: 1250 + 92a ≥ 7000 Question 4. Analyze what each part of 7x − 18 ≤ 32 means mathematically. Type below: ___________ Answer: x is the variable so it is the solution. 7x is the solution multiplied by 7. -18 means 7x is subtracted by 18. ≤ 32 means the result is no more than 32. Question 5. Write a real-world problem to represent 7x − 18 ≤ 32. Type below: ___________ Answer: A real-world problem could be: The temperature of a metal is currently at -18°C. A scientist will warm the metal at a rate of 7°C per hour until the temperature is 32°C. How many hours will it take to warm up the metal? Essential Question Check-In Question 6. Describe the steps you would follow to write a two-step inequality you can use to solve a real-world problem. Type below: ___________ Answer: The first step is to translate the words into an algebraic expression. The next step is to determine the target amount. The third step is to determine what inequality sign to use by determining if you need to be greater than, greater than or equal to, less than, less than, or equal to the target amount to write the inequality. Then solve the inequality sign, and target amount to write the inequality. Then solve the inequality for the unknown value. Finally, interpret the solution in the context of the problem. ### Independent Practice – Page No. 215 Question 7. Three friends earned more than$200 washing cars. They paid their parents $28 for supplies and divided the rest of the money equally. Write an inequality to find possible amounts each friend earned. Identify what your variable represents. Type below: ___________ Answer: 3x + 28 > 200 Explanation: Let x be the amount each friend received. Since there are 3 friends, then 3x is the amount of money they split evenly. The amount of money they split evenly was the amount left over after paying their parents$28.
Therefore 3x + 28 is the total amount they earned.
3x + 28 > 200  Question 10.
Due to fire laws, no more than 720 people may attend a performance at Metro Auditorium. The balcony holds 120 people. There are 32 rows on the ground floor, each with the same number of seats. Write an inequality to find the number of people that can sit in a ground-floor row if the balcony is full. Identify what your variable represents.
Type below:
___________

Answer: 32x + 120 ≤ 720

Explanation:
Let x represent the number of people that can sit in each ground floor row. then 32x is the total number of people sitting on the ground floor.
Since 120 people are sitting on the balcony, the total number of people is 32x + 120.

Question 11.
Liz earns a salary of $2,100 per month, plus a commission of 5% of her sales. She wants to earn at least$2,400 this month. Write an inequality to find amounts of sales that will meet her goal. Identify what your variable represents.
Type below:
___________

Answer: 2100 + 0.05x ≥ 2400

Explanation:
Let x represent the number of sales then 0.05x is the amount she earns in commission and 2100 + 0.05x is her total earnings.
2100 + 0.05x ≥ 2400

Question 12.
Lincoln Middle School plans to collect more than 2,000 cans of food in a food drive. So far, 668 cans have been collected. Write an inequality to find numbers of cans the school can collect on each of the final 7 days of the drive to meet this goal. Identify what your variable represents.
Type below:
___________

Answer: 7x + 668 > 2000

Explanation:
Let x represent the number of cans collected each day. Then 7x is the total number of cans collected on the final 7 days of the drive.
Since they have collected 668 cans already, the total number of cans collected is 7x + 668.
They want to collect more than 2000 cans, so the inequality is:
7x + 668 > 2000

Question 13.
Joanna joins a CD club. She pays $7 per month plus$10 for each CD that she orders. Write an inequality to find how many CDs she can purchase in a month if she spends no more than $100. Identify what your variable represents. Type below: ___________ Answer: 7 + 10x ≤ 100 Explanation: Let x represent the number of CDs then 10x is the total amount spent on CDs and 7 + 10x is the total purchase amount for the month. 7 + 10x ≤ 100 Question 14. Lionel wants to buy a belt that costs$22. He also wants to buy some shirts that are on sale for $17 each. He has$80. What inequality can you write to find the number of shirts he can buy? Identify what your variable represents.
Type below:
___________

Answer: 22 + 17x ≤ 80

Explanation:
Let x represent the number of shirts he can buy then 17x is the total cost of the shirts and 22 + 17x is the total cost
22 + 17x ≤ 80

### Page No. 216

Question 15.
Write a situation for 15x − 20 ≤ 130 and solve.
Type below:
___________

You are given in the inequality 15x − 20 ≤ 130 and need to write a situation that is represented by this inequality. A possible situation could be:
Type below:
__________

Let x represent the number of cars she must wash then 7x is the total amount she makes from washing cars. Since she is saving $25 of her earnings, 7x – 25 is the amount of earnings she will have to spend. 7x – 25 ≥ 65 Add 25 on both sides 7x – 25 + 25 ≥ 65 + 25 7x ≥ 90 Divide both sides by 7. x ≥ 90/7 ≈ 13 H.O.T. Focus on Higher Order Thinking Question 21. Analyze Relationships A compound inequality consists of two simple equalities joined by the word “and” or “or.” Graph the solution sets of each of these compound inequalities. a. x > 2 and x < 7 Type below: __________ Answer: Since the two inequalities are joined by “and”, we need to satisfy both inequalities. Therefore, we need all values greater than 2 but also we need values less than 7. Hence we can place a hollow dot on 2 and shade the line going to the right until another hollow dot on 7. Question 21. b. x < 2 or x > 7 Type below: __________ Answer: Since the two inequalities are joined by “or”, we need to satisfy either of the inequalities. Therefore, we need all values less than 2 but also we need to graph the values greater than 7. Hence we can place a hollow dot on 2 and shade the line going to the left and another hollow dot on 7 and shade the line going to the right. Question 21. c. Describe the solution set of the compound inequality x < 2 and x > 7. Type below: __________ Answer: The solution set for x > 2 and x < 7 should be 2 < x < 7. Question 21. d. Describe the solution set of the compound inequality x > 2 or x < 7. Type below: __________ Answer: The solution set for x < 2 and x > 7 sould be (-∞, 2) ∪ (7, ∞). Question 22. Communicate Mathematical Ideas Joseph used the problem-solving strategy Work Backward to solve the inequality 2n + 5 < 13. Shawnee solved the inequality using the algebraic method you used in this lesson. Compare the two methods. Type below: __________ Answer: Both involve using the same operations. The only difference is that working backward is done mostly mentally while algebraically is done on paper. It is easier to determine which direction the inequality is pointing when using the algebraic method. ### 7.1 Writing and Solving One-Step Inequalities – Page No. 223 Solve each inequality. Question 1. n + 7 < −3 Type below: __________ Answer: n < -10 Explanation: Subtract 7 on both sides n + 7 – 7 < −3 – 7 n < -10 Question 2. 5p ≥ −30 Type below: __________ Answer: p ≥ -6 Explanation: 5p ≥ −30 Divide by 5 on both sides p ≥ -6 Question 3. 14 < k + 11 Type below: __________ Answer: 3 < k Explanation: 14 < k + 11 Subtract 11 on both sides 14 – 11 < k + 11 – 11 3 < k Question 4. $$\frac{d}{-3}$$ ≤ −6 Type below: __________ Answer: d ≥ 18 Explanation: $$\frac{d}{-3}$$ ≤ −6 Multiply both sides by -3 remember to switch the inequality sign since you are multiplying both sides by a negative number. d ≥ 18 Question 5. c − 2.5 ≤ 2.5 Type below: __________ Answer: c ≤ 5 Explanation: c − 2.5 ≤ 2.5 Add 2.5 on both sides c − 2.5 + 2.5 ≤ 2.5 + 2.5 c ≤ 5 Question 6. 12 ≥ −3b Type below: __________ Answer: -4 ≤ b Explanation: 12 ≥ −3b Divide by -3 into both sides -4 ≤ b 7.2 Writing Two-Step Inequalities Question 8. During a scuba dive, Lainey descended to a point 20 feet below the ocean surface. She continued her descent at a rate of 20 feet per minute. Write an inequality you could solve to find the number of minutes she can continue to descend if she does not want to reach a point more than 100 feet below the ocean surface. Type below: __________ Answer: Let x represent the number of minutes. Since she is descending 20 feet per minute, then -20x represents her altitude. It is negative since descending means her altitude is decreasing. Since she started at 20 feet below the ocean surface, she started at -20 feet. It’s negative since an altitude below the ocean surface must be represented by a negative number. Her ending position is the sum of how far she has descended and her initial position so her ending position is -20 + (-20x) = -20 – 20x She doesn’t want to travel more than 100 feet below the ocean surface so she needs to be higher than -100 feet. The inequality is then -20 -20x ≥ -100. 7.3 Solving Two-Step Inequalities Solve. Question 9. 2s + 3 > 15 Type below: __________ Answer: s > 6 Explanation: 2s + 3 > 15 Subtract 3 on both sides 2s + 3 – 3 > 15 – 3 2s > 12 Divide by 2 on both sides s > 6 Question 10. −$$\frac{d}{12}$$ − 6 < 1 Type below: __________ Answer: d > -84 Explanation: −$$\frac{d}{12}$$ − 6 < 1 Add 6 on both sides −$$\frac{d}{12}$$ − 6 + 6 < 1 + 6 d > -84  Question 13. $$\frac{b}{9}$$ − 34 < −36 Type below: __________ Answer: b < -18 Explanation: $$\frac{b}{9}$$ − 34 < −36 Add 34 on both sides $$\frac{b}{9}$$ − 34 + 34 < −36 + 34 $$\frac{b}{9}$$ < -2 b < -18 Question 14. −2p + 12 > 8 Type below: __________ Answer: p < 2 Explanation: −2p + 12 > 8 Subtract 12 on both sides -2p + 12 – 12 > 8 – 12 -2p > -4 p < 2 Essential Question Question 15. How can you recognize whether a real-world situation should be represented by an equation or an inequality? Type below: __________ Answer: You use an equation when the situation involves finding an exact answer. You use an inequality when the solution can have more than one value. Problems that require the use of inequalities have phrases in them such as “at least”, “no more”, “at most” and “no less than”. ### Selected Response – Page No. 224 Question 1. Which graph models the solution of the inequality −6 ≤ −3x? Options: a. b. c. d. Answer: Dividing both sides of −6 ≤ −3x by -3 gives 2 ≥ x. Rewriting this so x is on the left side gives x ≤ 2. The graph must have a closed circle at 2 since the inequality has an equal sign and must be shaded to the left since its <. Thus the correct answer is option C. Question 2. A taxi cab costs$1.75 for the first mile and $0.75 for each additional mile. You have$20 to spend on your ride. Which inequality could be solved to find how many miles you can travel, if n is the number of additional miles?
Options:
a. 1.75n + 0.75 ≥ 20
b. 1.75n + 0.75 ≤ 20
c. 0.75n + 1.75 ≥ 20
d. 0.75n + 1.75 ≤ 20

Answer: 1.75n + 0.75 ≤ 20

Explanation:
Let n represent the number of additional miles. Then 0.75n is the cost of the additional miles which gives a total cost of 1.75 + 0.75n. You can spend a maximum of $20 so the inequality is ≤. Thus the correct answer is option B. Question 3. The inequality $$\frac{9}{5}$$C + 32 < −40 can be used to find Celsius temperatures that are less than -40° Fahrenheit. What is the solution to inequality? Options: a. C < 40 b. C < −$$\frac{40}{9}$$ c. C < −40 d. C < −$$\frac{72}{5}$$ Answer: C < −40 Explanation: $$\frac{9}{5}$$C + 32 < −40 Subtract 32 on both sides. $$\frac{9}{5}$$C + 32 – 32 < −40 – 32 $$\frac{9}{5}$$C < -72 c < -40 Thus the correct answer is option C. Question 4. The 30 members of a choir are trying to raise at least$1,500 to cover travel costs to a singing camp. They have already raised $600. Which inequality could you solve to find the average amounts each member can raise that will at least meet the goal? Options: a. 30x + 600 > 1,500 b. 30x + 600 ≥ 1,500 c. 30x + 600 < 1,500 d. 30x + 600 ≤ 1,500 Answer: 30x + 600 ≥ 1,500 Explanation: Given, The 30 members of a choir are trying to raise at least$1,500 to cover travel costs to a singing camp. They have already raised $600. Let x represent the average amount each member raises. There are 30 members so the members raise a combined amount of 30x. Since they have already raised$600, the total amount raised is 30x + 600 they need to raise at least $1500 so the inequality is ≥. Thus the correct answer is option B. Question 5. Which represents the solution for the inequality 3x − 7 > 5? Options: a. x < 4 b. x ≤ 4 c. x > 4 d. x ≥ 4 Answer: x > 4 Explanation: Add 7 on both sides 3x − 7 > 5 3x − 7 + 7> 5 + 7 3x > 12 x > 4 Thus the correct answer is option C. Question 6. Which inequality has the following graphed solution? Options: a. 3x + 8 ≤ 2 b. 4x + 12 < 4 c. 2x + 5 ≤ 1 d. 3x + 6 < 3 Answer: 4x + 12 < 4 Explanation: 4x + 12 < 4 Subtract 12 on both sides 4x + 12 – 12 < 4 – 12 4x < -8 x < -2 Thus the correct answer is option B. Question 7. Divide: −36 ÷ 6. Options: a. 30 b. 6 c. -6 d. -30 Answer: -6 Explanation: 6 divides 36 six times −36 ÷ 6 = -6 Thus the correct answer is option C. Question 8. Eleni bought 2 pounds of grapes at a cost of$3.49 per pound. She paid with a $10 bill. How much change did she get back? Options: a.$3.02
b. $4.51 c.$6.51
d. $6.98 Answer:$3.02

Explanation:
Given,
Eleni bought 2 pounds of grapes at a cost of $3.49 per pound. She paid with a$10 bill.
We have to find the total amount paid for the grapes
2 × 3.49 = 6.98
10 – 6.98 = 3.02
Thus the correct answer is option A.

Question 9.
In golf, the lower your score, the better. Negative scores are best of all. Teri scored +1 on each of the first three holes at a nine-hole miniature golf course. Her goal is a total score of -9 or better after she has completed the final six holes.
a. Let h represent the score Teri must average on each of the last six holes in order to meet her goal. Write a two-step inequality you can solve to find h.
Type below:
_____________

Explanation:
If h is her average score for the last 6 holes, then 6h is her total score for the last 6 holes.
She currently has a score of 3 so ger total score for all 9 holes is 3 + 6h.
She wants a score of -9 or better and since smaller scores are better, the inequality is ≤.

Question 9.
b. Solve the inequality.
Type below:
_____________

Explanation:
6h ≤ -12
Divide by 6 into both sides
h ≤ -2

### EXERCISES – Page No. 226

Simplify each expression.

Question 1.
$$\left(2 x+3 \frac{2}{5}\right)+\left(5 x-\frac{4}{5}\right)$$
Type below:
_____________

Answer: 7x + 2 $$\frac{3}{5}$$

Explanation:
We are given  the expression,
$$\left(2 x+3 \frac{2}{5}\right)+\left(5 x-\frac{4}{5}\right)$$
Group the like terms
(2x + 5x) + (3$$\frac{2}{5}$$ – $$\frac{4}{5}$$)
7x + 2 $$\frac{3}{5}$$  Question 4.
0.7(5a − 13p)
Type below:
_____________

Explanation:
0.7(5a − 13p)
0.7 × 5a – 0.7 × 13p
3.5a – 9.1p

Factor each expression.

Question 5.
8x + 56
Type below:
_____________

Explanation:
Since 56 ÷ 8 = 7 and 8 ÷ 8 = 1, factor out 8 from both terms
8x + 56 = 8(x + 7)

Question 6.
3x + 57
Type below:
_____________

Explanation:
Since 3 ÷ 3 = 1 and 57 ÷ 3 = 19 factors out 3 from both terms.
3x + 57 = 3(x + 19)

Question 7.
1.6 + y = −7.3
_______

Explanation:
subtract 1.6 on both sides
1.6 + y – 1.6 = -7.3 – 1.6
y = -8.9

Question 8.
−$$\frac{2}{3}$$ n = 12
_______

Explanation:
−$$\frac{2}{3}$$ n = 12
n = −$$\frac{3}{2}$$(12)
n = -3 × 6
n = -18  Question 11.
−5z + 4 = 34
_______

To plot a point, starting from 0, count the number of units going to the left or right.
Given equation is
−5z + 4 = 34
Subtract 4 on both sides
-5z + 4 – 4 = 34 – 4
-5z = 30
Divide both sides by -5
z = -6
To plot -6 on the number line, from 0, we move 6 units to the left.

### EXERCISES – Page No. 227

Question 1.
Prudie needs $90 or more to be able to take her family out to dinner. She has already saved$30 and wants to take her family out to eat in 4 days.
a. Suppose that Prudie earns the same each day. Write an inequality to find how much she needs to earn each day.
Type below:
___________

Let x be the amount she makes each day then 4x is the amount she will make in the 4 days before she takes her family out to eat and 4x + 30 is the total amount she will have saved.
4x + 30 ≥ 90

Question 1.
b. Suppose that Prudie earns $18 each day. Will she have enough money to take her family to dinner in 4 days? Explain. _______ Answer: 4(18) + 30 = 72 + 30 = 102 She will have saved$102 in total if she earns $18 each day so she will have enough money. Solve each inequality. Graph and check the solution. Question 2. 11 − 5y < −19 Type below: ___________ Answer: To graph inequalities, locate the number opposite the inequality variable on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution. From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >. First, solve the inequality: 11 − 5y < −19 Subtract 11 on both sides 11 – 5y – 11 < −19 – 11 -5y < -30 Divide by -5 into both sides y > 6 The number opposite to the variable is 6, we look for this in the number line. Since the inequality is >, we use a hollow dot and shade the line going to the right. Its graph would be like the one below: Question 3. 7x − 2 ≤ 61 Type below: ___________ Answer: To graph inequalities, locate the number opposite the inequality variable on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution. From here, shade the line going to the left if the inequality is either ≤ or < and shade the line going to the right if the inequality is either ≥ or >. First, solve the inequality: 7x − 2 ≤ 61 Add 2 on both sides 7x – 2 + 2 ≤ 61 + 2 7x ≤ 63 Divide by 7 into both sides x ≤ 9 The number opposite to the variable is 9, we look for this in the number line. Since the inequality is ≤, we use a closed dot and shade the line going to the left. Its graph would like the one below: ### Unit 3 Performance Tasks – Page No. 228 Question 1. Mechanical Engineer A mechanical engineer is testing the amount of force needed to make a spring stretch by a given amount. The force y is measured in units called Newtons, abbreviated N. The stretch x is measured in centimeters. Her results are shown in the graph. a. Write an equation for the line. Explain, using the graph and then using the equation, why the relationship is proportional. Type below: ___________ Answer: The graph is linear and passes through the origin so the relationship is proportional. find k by using the formula k = y/x where (x, y) is a point on the line then plug k into the equation of a line y = kx. Question 1. b. Identify the rate of change and the constant of proportionality. Type below: ___________ Answer: k = 8 Explanation: Observe part a the units N/cm since the units for y are N and the units for x are cms and the units for k must be the units for y divided by the units for x. 8 N/ cm k = 8 Question 1. c. What is the meaning of the constant of proportionality in the context of the problem? Type below: ___________ Answer: Since the rate of change is 8 N/ cm this means that for every 1 cm stretch in the spring, the force required in Newton increases by 8 N. Question 1. d. The engineer applies a force of 41.6 Newtons to the spring. Write and solve an equation to find the corresponding stretch in the spring. ______ cm Answer: y = 8x 41.6 = 8x x = 41.6/8 x = 5.2 cm Question 2. A math tutor charges$30 for a consultation, and then $25 per hour. An online tutoring service charges$30 per hour.
a. Does either service represent a proportional relationship? Explain.
Type below:
___________

The math tutor charges $30 initially but has a constant rate of$25 per hour after. This means that it still is a proportional relationship. The online tutoring charges a constant rate of $30 per hour and thus is also a proportional relationship. Question 2. b. Write an equation for the cost c of h hours of tutoring for either service. Which service charges less for 4 hours of tutoring? Show your work. Type below: ___________ Answer: Using y as the total cost and x as the number of hours, we can represent each tutoring service. For Math tutor, we can write this as y = 30 + 25x while for the online tutoring, we can write this as y = 30x. Substituting x = 4, we can see that: For Math tutor: y = 30 + 25x y = 30 + 25(4) y = 30 + 100 y = 130 For Online tutoring: y = 30x y = 30(4) y = 120 Therefore, the online tutoring service charges less at$120.

### Selected Response – Page No. 229 Question 2.
Timothy began the week with $35. He bought lunch at school, paying$2.25 for each meal. Let x be the number of meals he bought at school and y be the amount of money he had left at the end of the week. Which equation represents the relationship in the situation?
Options:
a. y = 2.25x + 35
b. y = 35 − 2.25x
c. x = 35 − 2.25y
d. y = 2.25x − 35

Answer: y = 35 − 2.25x

Explanation:
Let x be the number of meals he buys means 2.25x is the amount of money he has spent on meals. The money he has left is then 35 – 2.25x
The expression is y = 35 – 2.25x
Thus the correct answer is option B. Question 4.
Ramón’s toll pass account has a value of $32. Each time he uses the toll road,$1.25 is deducted from the account. When the value drops below $10, he must add value to the toll pass. Which inequality represents how many times Ramón can use the toll road without having to add value to the toll pass? Options: a. 10 − 1.25t ≥ 0 b. −1.25t + 32 < 10 c. 32 − 1.25t ≥ 10 d. 32 − 10t ≥ 1.25 Answer: 32 − 1.25t ≥ 10 Explanation: Let t represent the number of times he uses the toll road then 1.25t is the amount deducted from his account. the remaining balance is then 32 – 1.25t. Since his balance must be at least$10 for him to not have to add value, the inequality sign is ≥
Thus the correct answer is option C.

Question 5.
A taxi costs $1.65 for the first mile and$0.85 for each additional mile. Which equation could be solved to find the number x of additional miles traveled in a taxi given that the total cost of the trip is $20? Options: a. 1.65x + 0.85 = 20 b. 0.85x + 1.65 = 20 c. 1.65x − 0.85 = 20 d. 0.85x − 1.65 = 20 Answer: 0.85x + 1.65 = 20 Explanation: Let x be the number of additional miles means 0.85x is the cost of the additional miles the total cost is then 1.65 + 0.85x 1.65 + 0.85x = 20 Thus the correct answer is option B. Question 6. A sales tax of 6% is added to the price of an item. If Marisa buys an item, which expression indicates how much she will pay in all? Options: a. n + 0.06 b. 0.06n c. n + 0.06n d. 0.06 + 0.06n Answer: n + 0.06n Explanation: The total cost she will pay is the cost of the item n plus the cost of tax 0.06n. The expression is n + 0.06n Thus the correct answer is option C. Question 7. Which equation has the solution x = 12? Options: a. 4x + 3 = 45 b. 3x + 6 = 42 c. 2x − 5 = 29 d. 5x −8 = 68 Answer: 3x + 6 = 42 Explanation: a. 4x + 3 = 45 Substitute x = 12 in the above equation. 4(12) + 3 = 45 48 + 3 = 45 51 ≠ 45 b. 3x + 6 = 42 Substitute x = 12 in the above equation. 3(12) + 6 = 42 36 + 6 = 42 42 = 42 c. 2x − 5 = 29 Substitute x = 12 in the above equation. 2(12) – 5 = 29 24 – 5 = 29 19 ≠ 29 d. 5x −8 = 68 Substitute x = 12 in the above equation. 5(12) – 8 = 68 60 – 8 = 68 52 ≠ 68 Thus the correct answer is option B. Question 8. The 23 members of the school jazz band are trying to raise at least$1,800 to cover the cost of traveling to a competition. The members have already raised $750. Which inequality could you solve to find the amount that each member should raise to meet the goal? Options: a. 23x + 750 > 1,800 b. 23x + 750 ≥ 1,800 c. 23x + 750 < 1,800 d. 23x + 750 ≤ 1,800 Answer: 23x + 750 ≥ 1,800 Explanation: Let x represent the amount each member raises means 23x is the amount the members raise individually. The total amount raised is then 23x + 750 since they have already raised$750.
Since they are trying to raise at least $1800, the inequality is ≥ Thus the correct answer is option B. ### Page No. 230 Question 9. What is the solution of the inequality 2x − 9 < 7? Options: a. x < 8 b. x ≤ 8 c. x > 8 d. x ≥ 8 Answer: x < 8 Explanation: Given the inequality 2x − 9 < 7 Add 9 on both sides 2x – 9 + 9 < 7 + 9 2x < 16 Divide by 2 on both sides 2x/2 < 16/2 x < 8 Thus the correct answer is option A. Question 10. Which inequality has the solution n < 5? Options: a. 4n + 11 > −9 b. 4n + 11 < −9 c. −4n + 11 < −9 d. −4n + 11 > −9 Answer: −4n + 11 > −9 Explanation: Given the inequality n < 5 To graph inequalities, locate the number opposite the variable of the inequality on a number line. If the inequality is either a ≤ or a ≥, we use a closed dot, meaning the number is a solution as well. If the inequality is either a > or a <, use an open dot, indicating that the number is not a solution. a. 4n + 11 > −9 4n + 11 – 11 > -9 – 11 4n/4 > -20/4 n -5 b. 4n + 11 < −9 4n + 11 – 11 < -9 – 11 4n < -20 4n/4 < -20/4 n < -5 c. −4n + 11 < −9 -4n + 11 – 11 < -9 – 11 -4n < -20 -4n/-4 < -20/-4 n > 5 d. −4n + 11 > −9 -4n + 11 – 11 > -9 – 11 -4n > -20 -4n/-4 > -20/-4 n < 5 Thus the correct answer is option D. Question 11. Which inequality has the solution shown? Options: a. 3x + 5 < 2 b. 4x + 12 < 4 c. 2x + 5 ≤ 1 d. 3x + 6 ≤ 3 Answer: 3x + 6 ≤ 3 Explanation: The graph shows the inequality x ≤ -1 so the possible answers are C and D since A and B have < as the inequality signs. Solve C and D for x to see which one has x ≤ -1 as the solution. c. 2x + 5 ≤ 1 2x + 5 – 5 ≤ 1 – 5 2x ≤ -4 x ≤ -2 d. 3x + 6 ≤ 3 3x + 6 – 6 ≤ 3 – 6 3x ≤ -3 x ≤ -1 Thus the correct answer is option D. Question 12. On a 4 $$\frac{1}{2}$$ hour trip, Leslie drove $$\frac{2}{3}$$ of the time. For how many hours did Leslie drive? Options: a. 3 hours b. 3 $$\frac{1}{2}$$ hours c. 3 $$\frac{2}{3}$$ hours d. 3 $$\frac{5}{6}$$ hours Answer: 3 hours Explanation: Given that, On a 4 $$\frac{1}{2}$$ hour trip, Leslie drove $$\frac{2}{3}$$ of the time. Multiply the two fractions by first writing 4 $$\frac{1}{2}$$ as an improper fraction then cancel the 2s and then simplifying the division. 4 $$\frac{1}{2}$$($$\frac{2}{3}$$) = $$\frac{9}{2}$$ × $$\frac{2}{3}$$ = 3 Thus the correct answer is option A. Question 13. During a sale, the price of a sweater was changed from$20 to $16. What was the percent of decrease in the price of the sweater? Options: a. 4% b. 20% c. 25% d. 40% Answer: Mini-Task Question 14. Max wants to buy some shorts that are priced at$8 each. He decided to buy a pair of sneakers for $39, but the total cost of the shorts and the sneakers must be less than$75.
a. Write an inequality to find out how many pairs of shorts Max can buy.
Type below:
____________

Answer: 39 + 8x < 75

Explanation:
Let x be the number of shorts he buys then 8x is the total cost of the shorts and 8x + 39 is the total cost of the shorts and sneakers his total must be less than $75 so the inequality is <. 39 + 8x < 75 Question 14. b. Suppose that Max wants to buy 6 pairs of shorts. Will he have enough money? Explain. ______ Answer: No Explanation: Find the total amount he will spend buying 6 pairs of shorts this is more than the$75 he has so he will not have enough.
39 + 8(6) = 39 + 48 = 87

Question 14.
c. Solve the inequality to find the greatest number of pairs of shorts that Max can buy. Show your work.
______ pairs of shoes