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# Inequalities with the Distributive Property

## Includes combining like terms, variables on one side

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Solve Inequalities by Using the Distributive Property

Ms. Layne wants to build a rectangular deck in her back yard. She wants the length of the deck to be exactly 9 feet. She wants the perimeter of her deck to be, at most, 28 feet. The perimeter of any rectangle can be found by using the expression \begin{align*}P = 2(l + w)\end{align*}, where \begin{align*}l\end{align*} represents the length and \begin{align*}w\end{align*} represents the width. Write an inequality that could be used to represent \begin{align*}w\end{align*}, the possible widths, in feet, she could use for her deck. Would a deck with a width of 6 feet result in a deck with the perimeter she wants?

In this concept, you will solve inequalities by using the distributive property.

### Simplifying Inequalities

The inequalities that you will see in this concept involve parentheses. You can simplify an equation with parentheses by using the distributive property. You can do this with inequalities as well. Using the distributive property can help you to simplify an inequality so that it is easier to solve.

Solve for \begin{align*}q\end{align*}:

\begin{align*}-9(q+3)<45\end{align*}

First, apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by -9 and then add those products.

\begin{align*}\begin{array}{rcl} -9(q+3) &<& 45 \\ (-9 \times q)+(-9 \times 3) &<& 45 \\ -9q+(-27)&<& 45 \end{array}\end{align*}

Next, solve as you would solve any two-step inequality. Since -27 is added to \begin{align*}−9q\end{align*}, you can get \begin{align*}−9q\end{align*} by itself on one side of the inequality by subtracting -27 from both sides. Remember, subtracting -27 from a number is the same as adding its opposite, 27, to that number.

\begin{align*}\begin{array}{rcl} -9q+(-27)&<& 45 \\ -9q+(-27) -(-27) &<& 45 - (-27) \\ -9q+(-27+27)&<& 45 +27 \\ -9q &<& 72 \end{array}\end{align*}

Then, to get \begin{align*}q\end{align*} by itself on one side of the inequality, you need to divide both sides by -9. Since you are dividing both sides by a negative number, you need to reverse the inequality symbol.

\begin{align*}\begin{array}{rcl} -9q &<& 72 \\ \frac{-9q}{-9} &<& \frac{72}{-9}\\ q&>& -8 \end{array}\end{align*}

The answer is \begin{align*}q > -8\end{align*}.

Let’s try another example.

Solve for \begin{align*}x\end{align*}:

\begin{align*}\frac{1}{2}(x+4) \le 10\end{align*}

First, apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by \begin{align*}\frac{1}{2}\end{align*} and then add those products.

\begin{align*}\begin{array}{rcl} \frac{1}{2}(x+4) &\le & 10 \\ \left(\frac{1}{2} \times x \right)+\left(\frac{1}{2} \times 4 \right) &\le & 10 \\ \frac{1}{2}x+2 &\le & 10 \end{array}\end{align*}

Next, solve as you would solve any two-step inequality. Since 2 is added to \begin{align*}\frac{1}{2}x\end{align*}, you can get \begin{align*}\frac{1}{2}x\end{align*} by itself on one side of the inequality by subtracting 2 from both sides.

\begin{align*}\begin{array}{rcl} \frac{1}{2}x+2 &\le & 10 \\ \frac{1}{2}x+2 -2 &\le & 10 -2 \\ \frac{1}{2}x &\le & 8 \end{array}\end{align*}

Then, to get \begin{align*}x\end{align*} by itself on one side of the inequality, you need to multiply both sides by 2.

\begin{align*}\begin{array}{rcl} \frac{1}{2}x &\le & 8 \\ 2 \times \frac{1}{2}x &\le & 8 \times 2 \\ x &\le & 16 \end{array}\end{align*}

The answer is \begin{align*}x \le 16\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Ms. Layne's new deck.

First, consider part a. You know that the length is 9 feet, so substitute 9 for \begin{align*}l\end{align*} into the expression \begin{align*}2(l + w)\end{align*}. This expression represents the actual perimeter of the deck. Since she wants the perimeter to be “at most” 28 feet, you should use the “less than or equal to” \begin{align*}(\le)\end{align*} symbol.

\begin{align*}&P=2(l+w) \\ &28 \ge 2(9+w)\end{align*}

So, this problem can be represented by the inequality

\begin{align*}2(9+w) \le 28\end{align*}

Next, consider part b . To find all the possible values of \begin{align*}w\end{align*}, solve the inequality. First, apply the distributive property to the right side.

\begin{align*}\begin{array}{rcl} 2(9+w) &\le& 28 \\ (2 \times 9)+(2 \times w) &\le& 28 \\ 18+2w &\le & 28 \end{array}\end{align*}

Then, subtract 18 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} 18+2w &\le & 28 \\ 18-18+2w &\le& 28 -18 \\ 2w &\le & 10 \end{array}\end{align*}

Then, divide both sides of the inequality by 2.

\begin{align*} \begin{array}{rcl} 2w &\le & 10 \\ \frac{2w}{2} & \le & \frac{10}{2} \\ w & \le & 5 \end{array}\end{align*}

The value of \begin{align*}w\end{align*} must be less than or equal to 5. Since 6 is greater than, not less than, 5, it is not a possible value of \begin{align*}w\end{align*}. So, if she built her deck so it was 6 feet wide, it would have a larger perimeter than she wants.

#### Example 2

Solve for \begin{align*}w\end{align*}:

\begin{align*}-2(8+w)+18 < 28\end{align*}

First, apply the distributive property to the left side of the inequality. You can multiply each of the two numbers inside the parentheses by -2 and then add those products.

\begin{align*}\begin{array}{rcl} -2(8+w)+18 & < & 28 \\ (-2 \times 8)+(-2 \times w)+18 & < & 28 \\ -16+(-2w)+18 & < & 28 \end{array}\end{align*}

Next, add the like terms (-16 and 18) on the left side of the inequality. Using the commutative and associative properties to reorder the terms on the left side of the equation can make it easier to see how to do this.

\begin{align*}\begin{array}{rcl} -16+(-2w)+18 & < & 28 \\ (-16+18)-2w & < & 28 \\ 2-2w & < & 28 \end{array}\end{align*}

Then, subtract 2 from both sides of the inequality.

\begin{align*}\begin{array}{rcl} 2-2w & < & 28 \\ 2-2 -2w & < & 28 - 2\\ -2w & < & 26 \end{array}\end{align*}

Then, isolate the variable, \begin{align*}w\end{align*}, by dividing both sides of the inequality by -2. Since you are dividing both sides by a negative number, you need to reverse the inequality symbol.

\begin{align*}\begin{array}{rcl} -2w & < & 26 \\ \frac{-2w}{-2} & < & \frac{26}{-2}\\ w & > & -13 \end{array}\end{align*}

The answer is \begin{align*}w > -13\end{align*}.

#### Example 3

Solve for \begin{align*}x\end{align*}:

\begin{align*}-5(x+2)>15\end{align*}

First, apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by -5 and then add those products.

\begin{align*}\begin{array}{rcl} -5(x+2) & > & 15 \\ (-5 \times x)+(-5 \times 2) & > & 15 \\ -5x+(-10) & > & 15 \end{array}\end{align*}

Next, solve as you would solve any two-step inequality. Since -10 is added to \begin{align*}−5x\end{align*}, you can get \begin{align*}−5x\end{align*} by itself on one side of the inequality by subtracting -10 from both sides.

\begin{align*} \begin{array}{rcl} -5x+(-10) & > & 15 \\ -5x+(-10)-(-10) & > & 15-(-10) \\ -5x+(-10+10) & > & 15+10 \\ -5x & < & 25 \end{array}\end{align*}

Then, to get \begin{align*}x\end{align*} by itself on one side of the inequality, you need to divide both sides by -5. Since you are dividing both sides by a negative number, you need to reverse the inequality symbol.

\begin{align*}\begin{array}{rcl} -5x & > & 25 \\ \frac{-5x}{-5} & > & \frac{25}{-5} \\ x& < & -5 \end{array}\end{align*}

The answer is \begin{align*}x < -5\end{align*}.

#### Example 4

Solve for \begin{align*}x\end{align*}:

\begin{align*}6(x-4) \ge 24 \end{align*}

First, apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by 6 and then add those products.

\begin{align*}\begin{array}{rcl} 6(x-4) & \ge & 24 \\ (6 \times x)+(6 \times -4)& \ge & 24 \\ 6x+(-24) & \ge & 24 \end{array}\end{align*}

Next, subtract -24 from both sides.

\begin{align*}\begin{array}{rcl} 6x+(-24) & \ge & 24 \\ 6x+(-24)-(-24) & \ge & 24 - (-24) \\ 6x+(-24+24) & \ge & 24+24 \\ 6x & \ge & 48 \end{array}\end{align*}

Then, to get \begin{align*}x\end{align*} by itself on one side of the inequality, you need to divide both sides by 6.

\begin{align*} \begin{array}{rcl} 6x & \ge & 48 \\ \frac{6x}{6} & \ge & \frac{48}{6} \\ x & \ge & 8 \end{array}\end{align*}

The answer is \begin{align*}x \ge 8\end{align*}.

#### Example 5

Solve for \begin{align*}y\end{align*}:

\begin{align*}-2(y+2) \le 12\end{align*}

First, apply the distributive property to the left side of the inequality. Multiply each of the two numbers inside the parentheses by -2 and then add those products.

\begin{align*}\begin{array}{rcl} -2(y+3) & \le & 12 \\ (-2 \times y )+(-2 \times 3) & \le & 12\\ -2y+(-6) & \le & 12 \end{array}\end{align*}

Next, solve as you would solve any two-step inequality. Since -6 is added to \begin{align*}−2y\end{align*}, you can get \begin{align*}−2y\end{align*} by itself on one side of the inequality by subtracting -6 from both sides.

\begin{align*}\begin{array}{rcl} -2y+(-6) & \le & 12 \\ -2y+(-6)-(-6) & \le & 12 - (-6)\\ -2y+(-6+6) & \le & 12 +6 \\ -2y & \le & 18 \end{array}\end{align*}

Then, to get \begin{align*}y\end{align*} by itself on one side of the inequality, you need to divide both sides by -2. Remember to reverse the inequality symbol.

\begin{align*} \begin{array}{rcl} -2y & \le & 18 \\ \frac{-2y}{-2} & \le & \frac{18}{-2} \\ y & \ge & -9 \end{array}\end{align*}

The answer is \begin{align*}y \ge -9\end{align*}.

### Review

Solve each inequality.

1.  \begin{align*}3(x+4)>21\end{align*}
2.  \begin{align*}4(x-1)<8\end{align*}
3.  \begin{align*}5(y+7)<70\end{align*}
4.  \begin{align*}-4(x+2)>8\end{align*}
5.  \begin{align*}3(x-9) \ge 30\end{align*}
6.  \begin{align*}-2(y+4) \ge 16\end{align*}
7.  \begin{align*}5(x+2) \le 100\end{align*}
8.  \begin{align*}-2(y-3)+12y>16\end{align*}
9.  \begin{align*}4(x+2) -10x>38\end{align*}
10.  \begin{align*}3(x-2)+5x \le 42\end{align*}
11.  \begin{align*}-2(y+4)-2y>8\end{align*}
12.  \begin{align*}−5(x + 2) + 6(x− 2) \ge 10\end{align*}
13.  \begin{align*}3(x + 4) − 2(x + 1) > 5\end{align*}
14.  \begin{align*}−2(y− 4) + 8y + 2 < 16\end{align*}
15. \begin{align*}−8(x + 2) − 9x + 2x \ge 14\end{align*}

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### Vocabulary Language: English

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

like terms

Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.

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