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# 6.3: Multi-Step Inequalities

Difficulty Level: At Grade Created by: CK-12

The previous two lessons focused on one-step inequalities. Inequalities, like equations, can require several steps to isolate the variable. These inequalities are called multi-step inequalities. With the exception of the Multiplication/Division Property of Inequality, the process of solving multi-step inequalities is identical to solving multi-step equations.

Procedure to Solve an Inequality:

1. Remove any parentheses by using the Distributive Property.
2. Simplify each side of the inequality by combining like terms.
3. Isolate the \begin{align*}ax\end{align*} term. Use the Addition/Subtraction Property of Inequality to get the variable on one side of the inequality sign and the numerical values on the other.
4. Isolate the variable. Use the Multiplication/Division Property of Inequality to get the variable alone on one side of the inequality.
1. Remember to reverse the inequality sign if you are multiplying or dividing by a negative number.

Example: Solve for \begin{align*}w: \ 6x-5<10\end{align*}.

Solution: Begin by using the checklist above.

1. Parentheses? No

2. Like terms on the same side of inequality? No

3. Isolate the \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}6x-5+5<10+5\end{align*}

Simplify.

\begin{align*}6x<15\end{align*}

4. Isolate the variable using the Multiplication or Division Property.

\begin{align*}\frac{6x}{6} < \frac{15}{6} = x < \frac{5}{2}\end{align*}

5. Check your solution. Choose a number less than 2.5, say 0, and check using the original inequality.

\begin{align*}6(0)-5 &< 10?\\ -5 & < 10\end{align*}

Yes, the answer checks. \begin{align*}x < 2.5\end{align*}

Example: Solve for \begin{align*}x: \ -9x<-5x-15\end{align*}

Solution: Begin by using the checklist above.

1. Parentheses? No

2. Like terms on the same side of inequality? No

3. Isolate the \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}-9x+5x<-5x+5x-15\end{align*}

Simplify.

\begin{align*}-4x<-15\end{align*}

4. Isolate the variable using the Multiplication or Division Property.

\begin{align*}\frac{-4x}{-4} < \frac{-15}{-4}\end{align*}

Because the number you are dividing by is negative, you must reverse the inequality sign.

\begin{align*}x>\frac{15}{4} \rightarrow x > 3 \frac{3}{4}\end{align*}

5. Check your solution by choosing a number larger than 3.75, say 10.

\begin{align*}-9(10)& <-5(10)-15?\\ \checkmark \ -90 & <-65\end{align*}

Example: Solve for \begin{align*}x: \ 4x-2(3x-9) \le -4(2x-9)\end{align*}.

Solution: Begin by using the previous checklist.

1. Parentheses? Yes. Use the Distributive Property to clear the parentheses.

\begin{align*}4x+(-2)(3x)+(-2)(-9)\le-4(2x)+(-4)(-9)\end{align*}

Simplify.

\begin{align*}4x-6x+18 \le -8x+36\end{align*}

2. Like terms on the same side of inequality? Yes. Combine these.

\begin{align*}-2x+18\le-8x+36\end{align*}

3. Isolate the \begin{align*}ax\end{align*} term using the Addition Property.

\begin{align*}-2x+8x+18\le-8x+8x+36\end{align*}

Simplify.

\begin{align*}6x+18 & \le36\\ 6x+18-18 & \le36-18\\ 6x & \le18\end{align*}

4. Isolate the variable using the Multiplication or Division Property.

\begin{align*}\frac{6x}{6} \le \frac{18}{6} \rightarrow x \le 3\end{align*}

5. Check your solution by choosing a number less than 3, say –5.

\begin{align*}4(-5)-2(3 \cdot -5-9) & \le -4(2 \cdot -5-9)\\ \checkmark \ 28 & <76\end{align*}

## Identifying the Number of Solutions to an Inequality

Inequalities can have infinitely many solutions, no solutions, or a finite set of solutions. Most of the inequalities you have solved to this point have an infinite amount of solutions. By solving inequalities and using the context of a problem, you can determine the number of solutions an inequality may have.

Example: Find the solutions to \begin{align*}x-5>x+6\end{align*}.

Solution: Begin by isolating the variable using the Addition Property of Inequality.

\begin{align*}x-x-5>x-x+6\end{align*}

Simplify.

\begin{align*}-5>6\end{align*}

This is an untrue inequality. Negative five is never greater than six. Therefore, the inequality \begin{align*}x-5>x+6\end{align*} has no solutions.

Previously we looked at the following sentence: “The speed limit is 65 miles per hour.”

The algebraic sentence for this situation is: \begin{align*}s \le 65.\end{align*}

Example: Find the solutions to \begin{align*}s \le 65.\end{align*}

Solution: The speed at which you drive cannot be negative. Therefore, the set of possibilities using interval notation is [0, 65].

## Solving Real-World Inequalities

Example: In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?

Solution: The amount of subscriptions Leon needs is “at least” 120. Choose a variable to represent the varying quantity–the number of subscriptions, say \begin{align*}n\end{align*}. The inequality that represents the situation is \begin{align*}n+85 \ge 120\end{align*}.

Solve by isolating the variable \begin{align*}n\end{align*}: \begin{align*}n \ge 35\end{align*}.

Leon must sell 35 or more subscriptions to receive his bonus.

Example: The width of a rectangle is 12 inches. What must the length be if the perimeter is at least 180 inches? (Note: Diagram not drawn to scale.)

Solution: The perimeter is the sum of all the sides.

\begin{align*}12+12+x+x \ge 180\end{align*}

Simplify and solve for the variable \begin{align*}x\end{align*}:

\begin{align*}12+12+x+x & \ge 180 \rightarrow 24+2x \ge 180\\ 2x & \ge 156\\ x & \ge 78\end{align*}

The length of the rectangle must be 78 inches or larger.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

In 1 – 15, solve each of the inequalities and graph the solution set.

1. \begin{align*}6x-5 < 10\end{align*}
2. \begin{align*}-9x < -5x-15\end{align*}
3. \begin{align*}-\frac{9x}{5} \le 24\end{align*}
4. \begin{align*}\frac{9x}{5}-7 \ge -3x+12\end{align*}
5. \begin{align*}\frac{5x-1}{4} > -2 (x+5)\end{align*}
6. \begin{align*}4x+3 < -1\end{align*}
7. \begin{align*}2x < 7x - 36\end{align*}
8. \begin{align*}5x >8x + 27\end{align*}
9. \begin{align*}5 - x < 9 + x\end{align*}
10. \begin{align*}4-6x \le 2(2x+3)\end{align*}
11. \begin{align*}5(4x+3)\ge 9(x-2)-x\end{align*}
12. \begin{align*}2(2x-1)+3 < 5(x+3)-2x\end{align*}
13. \begin{align*}8x-5(4x+1) \ge -1+2(4x-3)\end{align*}
14. \begin{align*}2(7x-2)-3(x+2)< 4x-(3x+4)\end{align*}
15. \begin{align*}\frac{2}{3}x-\frac{1}{2}(4x-1) \ge x+2(x-3)\end{align*}
16. At the San Diego Zoo, you can either pay $22.75 for the entrance fee or$71 for the yearly pass, which entitles you to unlimited admission. At most, how many times can you enter the zoo for the $22.75 entrance fee before spending more than the cost of a yearly membership? 17. Proteek’s scores for four tests were 82, 95, 86, and 88. What will he have to score on his last test to average at least 90 for the term? 18. Raul is buying ties and he wants to spend$200 or less on his purchase. The ties he likes the best cost $50. How many ties could he purchase? 19. Virena's Scout Troop is trying to raise at least$650 this spring. How many boxes of cookies must they sell at \$4.50 per box in order to reach their goal?

Mixed Review

1. Solve: \begin{align*}10 \ge -5f.\end{align*}
2. Graph \begin{align*}y=-7\end{align*} on a coordinate plane.
3. Classify \begin{align*}\sqrt{5}\end{align*} using the real number hierarchy.
4. What are some problem-solving methods you have learned so far in this textbook? List one example for each method.
5. A circle has an area of \begin{align*}A=\pi r^2\end{align*}. What is the radius of a circle with area of \begin{align*}196\pi \ in^2\end{align*}?
6. Solve for \begin{align*}a: \frac{6}{a}=\frac{-22}{a+4}.\end{align*}

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