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<\/p>\nOpening Exercise:<\/p>\n
State whether each number sentence is true or false. If the number sentence is false, explain why.<\/p>\n
a. 4 + 5 > 9
\nAnswer:
\nFalse. 4 + 5 is not greater than 9.<\/p>\n
b. 36 = 18
\nAnswer:
\nTrue<\/p>\n
c. 32 > \\(\\frac{64}{4}\\)
\nAnswer:
\nTrue<\/p>\n
d. 78 – 15 < 68
\nAnswer:
\nTrue<\/p>\n
e. 22 \u2265 11 + 12
\nAnswer:
\nFalse. 22 is not greater than or equal to 23.<\/p>\n
Example 1:<\/p>\n
Write true or false if the number substituted for g results in a true or false number sentence.<\/p>\n
<\/p>\n
Answer:<\/p>\n
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<\/p>\n
Example 2:<\/p>\n
State when the following equations\/inequalities will be true and when they will be false.<\/p>\n
a. r + 15 = 25
\nAnswer:
\n\u2192 Can you think of a number that will make this equation true?
\nYes. Substituting 10 for r will make a true number sentence.<\/p>\n
\u2192 Is 10 the only number that results in a true number sentence? Why or why not?
\nYes. There is only one value that, if substituted, will result in a true number sentence. There is only one number that can be added to 15 to get exactly 25.<\/p>\n
\u2192 What will make the number sentence false?
\nAny number that is not 10 will result in a false number sentence.<\/p>\n
\u2192 If we look back to the original questions, how can we state when the equation will be true? False?
\nThe equation is true when the value substituted for r is 10 and false when the value of r is any other number.<\/p>\n
b. 6 – d > 0
\nAnswer:
\n\u2192 If we wanted 6 – d to equal 0, what would the value of d have to be? Why?
\nThe value of d would have to be 6 because 6 – 6 = 0.<\/p>\n
\u2192 Will substituting 6 for d result in a true number sentence? Why or why not?
\nIf d has a value of 6, then the resulting number sentence would not be true because the left side has to be greater than 0, not equal to 0.<\/p>\n
\u2192 How about substituting 5 for d? 4? 3? 2?
\nYes. Substituting any of these numbers for d into the inequality results in true number sentences.<\/p>\n
\u2192 What values can we substitute for d in order for the resulting number sentence to be true?
\nThe inequality is true for any value of d that is less than 6.<\/p>\n
\u2192 What values for d would make the resulting number sentence false?
\nThe inequality is false for any value of d that is greater than or equal to 6.<\/p>\n
\u2192 Let\u2019s take a look at a number line and see why these statements make sense.<\/p>\n
Display a number line on the board. Label the number line as shown below.<\/p>\n
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<\/p>\n
\u2192 Let\u2019s begin at 6. If I were to subtract 1 from 6, where would that place be on the number line?
\n5<\/p>\n
\u2192 So, if we substitute 1 for d, then 6 – 1 = 5, and the resulting number sentence is true. How about if I subtracted 2 from 6? Would our number sentence be true for the value 2?
\nYes<\/p>\n
\u2192 What if I subtracted 6 from the 6 on the number line? Where would that be on the number line?
\n0<\/p>\n
\u2192 So, if we substitute 6 for d, will the resulting number sentence be true or false?
\nFalse<\/p>\n
\u2192 Let\u2019s try one more. We have already determined that any number greater than or equal to 6 will result in a false number sentence. Let\u2019s try a number greater than 6. Let\u2019s try the number 7.<\/p>\n
\u2192 Start with the 6 on the number line. If we were to subtract 7, in which direction on the number line would we
\nmove?
\nTo the left<\/p>\n
\u2192 And how many times will we move to the left?
\n7<\/p>\n
\u2192 Model beginning at 6 on the number line, and move a finger, or draw the unit skips, while continually moving to the left on the number line 7 times.<\/p>\n
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<\/p>\n
\u2192 So, it seems we have ended up at a place to the left of O. What number is represented by this position?
\n-1<\/p>\n
c. \\(\\frac{1}{2}\\)f = 15
\nAnswer:
\nThe equation is true when the value substituted for fis 30 (f = 30) and false when the value off is any other number (f \u2260 30).<\/p>\n
d. \\(\\frac{y}{3}\\) < 10
\nAnswer:
\nThe inequality is true for any value of y that is less than 30 (y < 30) and false when the value of y is greater than or equal to 30 (y \u2265 30).<\/p>\n
e. 7g \u2265 42
\nAnswer:
\nThe inequality is true for any value of g that is greater than or equal to 6 (g \u2265 6) and false when the value of g is less than (g < 6).<\/p>\n
f. a – 8 \u2264 15
\nAnswer:
\nThe inequality is true for any value of a that is less than or equal to 23 (a \u2264 23) and false when the value of a is greater than 23 (a > 23).<\/p>\n
Exercises:<\/p>\n
Complete the following problems in pairs. State when the following equations and inequalities will be true and when they will be fake.<\/p>\n
Exercise 1.
\n15c > 45
\nAnswer:
\nThe inequality is true for any value of c that is greater than 3 (c > 3) and false when the value of c is less than or equal to (c \u2264 3).<\/p>\n
Exercise 2.
\n25 = d – 10
\nAnswer:
\nThe equation is true when the value of d is 35 (d = 35) and false when the value of d is any other number (d \u2260 35).<\/p>\n
Exercise 3.
\n56 \u2265 2e
\nAnswer:
\nThe inequality is true for any value of e that is less than or equal to 28 (e \u2264 28) and false when the value of e is greater than 8 (e > 28).<\/p>\n
Exercise 4.
\n\\(\\frac{h}{5}\\) \u2265 12
\nAnswer:
\nThe inequality is true for any value of h that is greater than or equal to 60 (h \u2265 60) and false when the value of h is less than (h < 60).<\/p>\n
Exercise 5.
\n45 > h + 29
\nAnswer:
\nThe inequality is true for any value of h that is less than 16 (h < 16) and false when the value of h is greater than or
\nequal to 16 (h \u2265 16).<\/p>\n
Exercise 6.
\n4a \u2264 16
\nAnswer:
\nThe inequality is true for any value of a that is less than or equal to 4 (a \u2264 4) and false when the value of a is greater than (a > 4).<\/p>\n
Exercise 7.
\n3x = 24
\nAnswer:
\nThe equation is true when the value of x is 8 (x = 8) and false when the value of x is any other number (x \u2260 8).<\/p>\n
Identify all equality and inequality signs that can be placed into the blank to make a true number sentence.<\/p>\n
Exercise 8.
\n15 + 9 ___ 24
\nAnswer:
\n= or \u2265 or \u2264<\/p>\n
Exercise 9.
\n8.7 ___ 50
\nAnswer:
\n> or \u2265<\/p>\n
Exercise 10.
\n\\(\\frac{15}{2}\\) ___ 10
\nAnswer:
\n< or \u2264<\/p>\n
Exercise 11.
\n34 ___ 17 . 2
\nAnswer:
\n= or \u2265 or \u2264<\/p>\n
Exercise 12.
\n18 ___ 24.5 – 6
\nAnswer:
\n< or \u2264<\/p>\n
State when the following equations and inequalities will be true and when they will be false.<\/p>\n
Question 1.
\n36 = 9k
\nAnswer:
\nThe equation is true when the value of k is 4 and false when the value of k is any number other than 4.
\nOR
\nThe equation is true when k = 4 and false when k \u2260 4.<\/p>\n
Question 2.
\n67 > f – 15
\nAnswer:
\nThe inequality is true for any value off that is less than 82 and false when the value of f is greater than or equal to 82.
\nOR
\nThe inequality is true when f < 82 and false when f \u2265 82.<\/p>\n
Question 3.
\n\\(\\frac{v}{9}\\) = 3
\nAnswer:
\nThe equation is true when the value of v is 27 and false when the value of v is any number other than 27.
\nOR
\nThe equation is true when v = 27 and false when v \u2260 27.<\/p>\n
Question 4.
\n10 + b > 42
\nAnswer:
\nThe inequality is true for any value of b that is greater than 32 and false when the value of b is less than or equal to 32.
\nOR
\nThe inequality is true when b > 32 and false when b \u2264 32.<\/p>\n
Question 5.
\nd – 8 \u2265 35
\nAnswer:
\nThe inequality is true for any value of d that is greater than or equal to 43 and false when the value of d is less than 43
\nOR
\nThe inequality is true when d \u2265 43 and false when d <43.<\/p>\n
Question 6.
\n32f < 64
\nAnswer:
\nThe inequality is true for any value of f that is less than 2 and false when the value off is greater than or equal to 2.
\nOR
\nThe inequality is true when f < 2 and false when f \u2265 2.<\/p>\n
Question 7.
\n10 – h \u2264 7
\nAnswer:
\nThe inequality is true for any value of h that is greater than or equal to 3 and false when the value of h is less than 3.
\nOR
\nThe inequality is true when h \u2265 3 and false when h < 3.<\/p>\n
Question 8.
\n42 + 8 \u2265 g
\nAnswer:
\nThe inequality is true for any value of g that is less than or equal to 50 and false when the value of g is greater than 50.
\nOR
\nThe inequality is true when g < 50 and false when g > 50.<\/p>\n
Question 9.
\n\\(\\frac{m}{3}\\) = 14
\nAnswer:
\nThe equation is true when the value of m is 42 and false when the value of m is any number other than 42.
\nOR
\nThe equation is true when m = 42 and false when m \u2260 42.<\/p>\n
State when the following equations and inequalities will be true and when they will be false.<\/p>\n
Question 1.
\n5g > 45
\nAnswer:
\nThe inequality is true for any value of g that is greater than 9 and false when the value of g is less than or equal to 9.
\nOR
\nThe inequality is true when g > 9 and false when g \u2264 9.<\/p>\n
Question 2.
\n14 = 5 + k
\nAnswer:
\nThe equation is true when the value of k is 9 and false when the value of k is any other number.
\nOR
\nThe equation is true when k = 9 and false when k \u2260 9.<\/p>\n
Question 3.
\n26 – w < 12
\nAnswer:
\nThe inequality is true for any value of w that is greater than 14 and false when the value of w is less than or equal to 14.
\nOR
\nThe inequality is true when w > 14 and false when w < 14.<\/p>\n
Question 4.
\n32 \u2264 a + 8
\nAnswer:
\nThe inequality is true for any value of a that is greater than or equal to 24 and false when the value of a is less than 24.
\nOR
\nThe inequality is true when a \u2265 24 and false when a < 24.<\/p>\n
Question 5.
\n2 . h \u2264 16
\nAnswer:
\nThe inequality is true for any value of h that is less than or equal to 8 and false when the value of h is greater than 8.
\nOR
\nThe inequality is true when h < 8 and false when h > 8.<\/p>\n","protected":false},"excerpt":{"rendered":"
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