3.5 Solve Equations Using Integers; The Division Property of Equality

Before you get started, take this readiness quiz.

$\text{Evaluate} x + 4 \text{when} x = −4 .$
If you missed this problem, review the reference.

$\text{Solve}: y - 6 = 10 .$
If you missed this problem, review the reference.

Translate into an algebraic expression $5$ less than $x .$
If you missed this problem, review the reference.

Determine whether each of the following is a solution of $2 x - 8 = −14 :$

1. ⓐ $x = −11$
2. ⓑ $x = 11$
3. ⓒ $x = −3$

Determine whether each of the following is a solution of $2 y + 3 = −11 :$

1. ⓐ $y = 4$
2. ⓑ $y = −4$
3. ⓒ $y = −7$

Solve:

$y + 11 = 7$

Solve:

$y + 15 = −4$

Solve:

$a - 2 = −8$

Solve:

$n - 4 = −8$

Solve:

$8 a = 56$

Solve:

$11 n = 121$

Solve:

$−8 p = 96$

Solve:

$−12 m = 108$

Translate and solve: five more than $x$ is equal to $−3 .$

Translate and solve:

Seven more than $x$ is equal to $−2$.

Translate and solve:

$\text{Eleven more than} y \text{is equal to 2}.$

Translate and solve: the difference of $n$ and $6$ is $−10 .$

Translate and solve:

The difference of $p$ and $2$ is $−4$.

Translate and solve:

The difference of $q$ and $7$ is $−3$.

Translate and solve: the number $108$ is the product of $−9$ and $y .$

Translate and solve:

The number $132$ is the product of $−12$ and $y$.

Translate and solve:

The number $117$ is the product of $−13$ and $z$.

Determine whether each number is a solution of the given equation.

$4 x - 2 = 6$

1. ⓐ $x = −2$
2. ⓑ $x = −1$
3. ⓒ $x = 2$

Determine whether each number is a solution of the given equation.

$9 a + 27 = −63$

1. ⓐ $a = 6$
2. ⓑ $a = −6$
3. ⓒ $a = −10$

Solve for the unknown.

$n + 12 = 5$

Solve for the unknown.

$p + 9 = −8$

Solve for the unknown.

$u - 3 = −7$

Solve for the unknown.

$h - 10 = −4$

Solve for the unknown.

$x + \left(−2\right) = −18$

Solve for the unknown.

$r - \left(−5\right) = −9$

Solve each equation using the division property of equality and check the solution.

$5 x = 45$

Solve each equation using the division property of equality and check the solution.

$−7 c = 56$

Solve each equation using the division property of equality and check the solution.

$−14 p = −42$

Solve each equation using the division property of equality and check the solution.

$−120 = 10 q$

Solve each equation using the division property of equality and check the solution.

$24 x = 480$

Solve each equation using the division property of equality and check the solution.

$−3 z = 0$

Translate and solve.

Four more than $n$ is equal to 1.

Translate and solve.

The sum of eight and $p$ is $−3$.

Translate and solve.

The difference of $a$ and three is $−14$.

Translate and solve.

The number −42 is the product of −7 and $x$.

Translate and solve.

The product of -15 and $f$ is 75.

Translate and solve.

−6 plus $c$ is equal to 4.

Translate and solve.

Nine less than $m$ is −4.

Solve.

1. ⓐ $x + 2 = 10$
2. ⓑ $2 x = 10$

Solve.

1. ⓐ $−3 p = 27$
2. ⓑ $p - 3 = 27$

Solve.

$a - 4 = 16$

Solve.

$−8 m = −56$

Solve.

$−39 = u + 13$

Solve.

$11 r = −99$

Solve.

$100 = 20 d$

Solve.

$−49 = x - 7$

Cookie packaging A package of $\text{51 cookies}$ has $3$ equal rows of cookies. Find the number of cookies in each row, $c ,$ by solving the equation $3 c = 51 .$

Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation $3 x = 15 ?$ Explain why or why not.

Frida started to solve the equation $−3 x = 36$ by adding $3$ to both sides. Explain why Frida’s method will not solve the equation.

Order each of the following pairs of numbers, using $<$ or $>.$

$4 __ 8$

Order each of the following pairs of numbers, using $<$ or $>.$

$−5 __ −10$

Order each of the following pairs of numbers, using $<$ or $>.$

$2 __ −7$

Find the opposite of each number.

$6$

Find the opposite of each number.

$−4$

Simplify.

1. ⓐ $- \left(8\right)$
2. ⓑ $- \left(−8\right)$

Evaluate.

$- x , \text{when}$

1. ⓐ $x = 32$
2. ⓑ $x = −32$

Simplify.

$\left|−21\right|$

Simplify.

$\left|36\right|$

Simplify.

$\left|0\right|$

Evaluate.

$\left|x\right| \text{when} x = −14$

Evaluate.

$- \left|- y\right| \text{when} y = 33$

Fill in $< , > , \text{or} =$ for each of the following pairs of numbers.

$- \left|−4\right| __ 4$

Fill in $< , > , \text{or} =$ for each of the following pairs of numbers.

$- \left|−6\right| __ −6$

Simplify.

$- \left(−55\right) \text{and} - \left|−55\right|$

Simplify.

$\left|12 - 5\right|$

Simplify.

$6 \left|−9\right|$

Simplify.

$\left|9 - 3\right| - \left|5 - 12\right|$

Translate each of the following phrases into expressions with positive or negative numbers.

the opposite of $16$

Translate each of the following phrases into expressions with positive or negative numbers.

negative $3$

Translate each of the following phrases into expressions with positive or negative numbers.

a temperature of $10$ below zero

Model the following to find the sum.

$3 + 7$

Model the following to find the sum.

$5 + \left(−4\right)$

Simplify each expression.

$14 + 82$

Simplify each expression.

$−75 + 25$

Simplify each expression.

$11 + \left(−15\right) + 3$

Simplify each expression.

$−3 + 6 \left(−1 + 5\right)$

Evaluate each expression.

$n + 4 \text{when}$

1. ⓐ $n = −1$
2. ⓑ $n = −20$

Evaluate each expression.

$\left(x + y\right)^{3} \text{when} x = −4 , y = 1$

Translate each phrase into an algebraic expression and then simplify.

$\text{the sum of }−\text{8 and 2}$

Translate each phrase into an algebraic expression and then simplify.

$\text{10 more than the sum of }−\text{5 and }−\text{6}$

Solve.

Temperature On Monday, the high temperature in Denver was $−\text{4 degrees}.$ Tuesday’s high temperature was $\text{20 degrees}$ more. What was the high temperature on Tuesday?

Simplify each expression.

$24 - 16$

Simplify each expression.

$−31 - 7$

Simplify each expression.

$−52 - \left(−17\right) - 23$

Simplify each expression.

$\left(1 - 7\right) - \left(3 - 8\right)$

Evaluate each expression.

$x - 7 \text{when}$

1. ⓐ $x = 5$
2. ⓑ $x = −4$

Evaluate each expression.

$2 n^{2} - n + 5 \text{when} n = −4$

Translate each phrase into an algebraic expression and then simplify.

the difference of $−12 \text{and} 5$

Solve the given applications.

Temperature One morning the temperature in Bangor, Maine was $\text{18 degrees}.$ By afternoon, it had dropped $\text{20 degrees}.$ What was the afternoon temperature?

Multiply.

$−9 \cdot 4$

Multiply.

$\left(−11\right) \left(−11\right)$

Divide.

$56 \div \left(−8\right)$

Divide.

$−96 \div 12$

Divide.

$45 \div \left(−1\right)$

Simplify each expression.

$5 \left(−9\right) - 3 \left(−12\right)$

Simplify each expression.

$- 3^{4}$

Simplify each expression.

$42 - 4 \left(6 - 9\right)$

Simplify each expression.

$−2 \left(−18\right) \div 9$

Evaluate each expression.

$7 x - 3 \text{when} x = −9$

Evaluate each expression.

$5 a + 8 b \text{when} a = −2 , b = −6$

Translate to an algebraic expression and simplify if possible.

the product of $−12$ and $6$

Determine whether each number is a solution of the given equation.

$5 x - 10 = −35$

1. ⓐ $x = −9$
2. ⓑ $x = −5$
3. ⓒ $x = 5$

Solve.

$a + 14 = 2$

Solve.

$c + \left(−10\right) = −17$

Solve each equation using the division property of equality and check the solution.

$8 p = 72$

Solve each equation using the division property of equality and check the solution.

$−16 r = −64$

Translate and solve.

$\text{The product of }−\text{6 and} y \text{is} −42$

Translate and solve.

Four more than $m$ is $−48 .$

Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

Compare the numbers, using $< \text{or} > \text{or} =.$

1. ⓐ $−6 __ 3$
2. ⓑ $−1 __ −4$

Find the opposite of each number.

1. ⓐ $−7$
2. ⓑ $8$

Simplify.

$\left|4 - 9\right|$

Simplify.

$−15 + \left(−12\right)$

Simplify.

$10 - \left(5 - 6\right)$

Simplify.

$−6 \left(−9\right)$

Simplify.

$\left(−2\right)^{3}$

Simplify.

$16 −3 \left(5 −7\right)$

Evaluate.

$35 - a \text{when} a = −4$

Evaluate.

$3 m - 2 n \text{when} m = 6 , n = −8$

Translate each phrase into an algebraic expression and then simplify, if possible.

the difference of −7 and −4

Solve.

Early one morning, the temperature in Syracuse was $−8 °\text{F}.$ By noon, it had risen $\text{12}°.$ What was the temperature at noon?

Solve.

$n + 6 = 5$

Solve.

$−9 r = −54$

Translate and solve.

$\text{Eight less than} y \text{is }−\text{32}.$