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Study Guides > Finite Math

Reading: Multi-Step Inequalities

In the last two sections, we considered very simple inequalities which required one-step to obtain the solution. However, most inequalities require several steps to arrive at the solution. As with solving equations, we must use the order of operations to find the correct solution. In addition remember that when we multiply or divide the inequality by a negative number the direction of the inequality changes. The general procedure for solving multi-step inequalities is as follows.
  1. Clear parenthesis on both sides of the inequality and collect like terms.
  2. Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
  3. Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality if you multiply or divide by a negative number.

Solve a Two-Step Inequality

Example 1

Solve each of the following inequalities and graph the solution set.
  1. 6x − 5 < 10
  2. −9x < −5x − 15
  3. [latex]\displaystyle-\frac{{{9}{x}}}{{5}}[/latex]$latex \le $ 24

Solution

  1. 6x – 5 < 10Add 5 to both sides: 6 x –5 + 5 < 10 +5Simplify: 6 x < 15Divide both sides by 6: [latex-display]\displaystyle\frac{{{6}{x}}}{{6}}\lt\frac{{15}}{{6}}[/latex-display] Simplify: [latex]\displaystyle{x}\lt\frac{{5}}{{2}}[/latex] A number line with points marked from negative 10 to positive 10. There is an empty dot at point 2 and a half with a line extending past the number line to negative infinity.
  2. –9x < –5x – 15 Add 5x to both sides: –9x + 5x < –5x + 5x – 15 Simplify: –4x < –15
Divide both sides by –4: [latex-display]\displaystyle\frac{{-{4}{x}}}{{-{{4}}}}\gt-\frac{{15}}{{-{{4}}}}[/latex-display] Inequality sign is flipped Simplify: [latex-display]\displaystyle{x}\gt\frac{{15}}{{4}}[/latex-display] A number line with points marked from negative 10 to positive 10. There is an empty dot at point 3 and a three-fourths with a line extending past the number line to infinity. 3. [latex]\displaystyle-\frac{{{9}{x}}}{{5}}[/latex]$latex \le $ 24 Multiply both sides by 5: [latex-display]\displaystyle{5}\frac{{-{9}{x}}}{{5}}\le{24}({5})[/latex-display] Simplify: –9 x $latex \le $120 Divide both sides by –9: [latex-display]\displaystyle\frac{{-{9}{x}}}{{-{{9}}}}\ge\frac{{120}}{{-{{9}}}}[/latex-display] Inequality sign is flipped. Simplify: [latex-display]\displaystyle{x}\ge-\frac{{40}}{{3}}[/latex-display] A number line with points marked from negative 15 to positive 5. There is an empty dot at point negative 13 and a one-third with a line extending past the number line to infinity.

Solve a Multi-Step Inequality

Example 2

Each of the following inequalities and graph the solution set.
  1. [latex]\displaystyle\frac{{{9}{x}}}{{5}}-{7}\ge-{3}{x}+{12}[/latex]
  2. −25x+12$latex \le $ −10x−12

Solutions

[latex-display]\displaystyle\frac{{{9}{x}}}{{5}}-{7}\ge-{3}{x}+{12}[/latex-display]
  1. Add 3x to both sides: [latex]\displaystyle\frac{{{9}{x}}}{{5}}+{3}{x}-{7}\ge-{3}{x}+{3}{x}+{12}[/latex]Simplify: [latex]\displaystyle\frac{{{24}{x}}}{{5}}-{7}\ge{12}[/latex]Add 7 to both sides: [latex-display]\displaystyle\frac{{{24}{x}}}{{5}}-{7}+{7}\ge{12}+{7}[/latex-display] Simplify: [latex-display]\displaystyle\frac{{{24}{x}}}{{5}}\ge{19}[/latex-display] Multiply both sides by 5: [latex-display]\displaystyle{5}\frac{{{24}{x}}}{{5}}\ge{19}({5})[/latex-display] Simplify: [latex-display]\displaystyle{24}{x}\ge{95}[/latex-display] Divide both sides by 24: [latex-display]\displaystyle\frac{{{24}{x}}}{{24}}\ge\frac{{95}}{{24}}[/latex-display] Simplify: A number line with points marked from negative 10 to positive 10. There is a solid dot at point 3.95 and a half with a line extending past the number line to infinity.
  2. –25x + 12$latex \le $ –10x – 12Add 10x to both sides: –25x + 10x + 12$latex \le $ –10x + 10x – 12Simplify: –15x + 12 $latex \le $ –12Subtract 12 from both sides: –15x + 12 – 12$latex \le $ –12 – 12 Simplify: –15x $latex \le $ –24 Divide both sides by –15: [latex]\displaystyle\frac{{-{15}{x}}}{{-{{15}}}}\ge\frac{{-{24}}}{{-{{15}}}}[/latex] Inequality sign is flipped. Simplify: [latex]\displaystyle{x}\ge\frac{{8}}{{5}}[/latex] A number line with points marked from negative 10 to positive 10. There is an empty dot at point 1 and a three-fifths with a line extending past the number line to infinity.

Example 3

Solve the following inequalities.
  1. 4x − 2(3x − 9)$latex \le $ −4(2x − 9)
  2. [latex]\displaystyle\frac{{{5}{x}-{1}}}{{4}}[/latex]> −2(x + 5)

Solutions

  1. 4x − 2(3x − 9)$latex \le $ −4(2x − 9)
Simplify parentheses: 4 – 6 x + 18$latex \le $ –8x + 36 Collect like terms: –2 x + 18$latex \le $ –8x + 36 Add 8x to both sides: –2x + 8x + 18$latex \le $–8x + 8x + 36 Simplify: 6 x + 18$latex \le $ 36 Subtract 18 from both sides: 6 x + 18 –18$latex \le $ 36 –18 Simplify: 6 x $latex \le $ 18 Divide both sides by 6: [latex-display]\displaystyle\frac{{{6}{x}}}{{6}}\le\frac{{18}}{{6}}[/latex-display] Simplify: x $latex \le $ 3
  1. [latex]\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{({x}+{5})}[/latex]Simplify parenthesis: [latex]\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{x}-{10}[/latex]Multiply both sides by 4: [latex]\displaystyle{4}\frac{{{5}{x}-{1}}}{{4}}\gt{4}{(-{2}{x}-{10})}[/latex]Simplify: 5 x –1 > –8x –40 Add 8x to both sides: 5x +8x –1 > 8x + 8x – 40 Simplify: 13x –1 > – 40 Add 1 to both sides: 13x –1 + 1 > – 40 + 1 Simplify: 13 x > – 39 Divide both sides by 13: [latex]\displaystyle\frac{{{13}{x}}}{{13}}\gt-\frac{{39}}{{13}}[/latex] Simplify: x > –3

Identify the Number of Solutions of an Inequality

Inequalities can have:
  • A set that has an infinite number of solutions.
  • No solutions
  • A set that has a discrete number of solutions.

Infinite Number of Solutions

The inequalities we have solved so far all have an infinite number of solutions. In the last example, we saw that the inequality [latex]\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{({x}+{5})}[/latex]has the solution x > −3 This solution says that all real numbers greater than −3 make this inequality true. You can see that the solution to this problem is an infinite set of numbers.

No solutions

Consider the inequality x − 5 > x + 6 This simplifies to −5 > 6 This statements is not true for any value of x. We say that this inequality has no solution.

Discrete solutions

So far we have assumed that the variables in our inequalities are real numbers. However, in many real life situations we are trying to solve for variables that represent integer quantities, such as number of people, number of cars or number of ties.

Example 4

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?

Solution

Let x = the number of ties Raul purchases. We can write an inequality that describes the purchase amount using the formula. (number of ties) × (price of a tie) le $200 or 50x le 200 We simplify our answer. x$latex \le $ 4 This solution says that Raul bought four or less ties. Since ties are discrete objects, the solution set consists of five numbers {0,1,2,3,4}.

Solve Real-World Problems Using Inequalities

Sometimes solving a word problem involves using an inequality

Example 5

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

Step 1

We know that Leon sold 85 subscriptions and he must sell at least 120 subscriptions. We want to know the least amount of subscriptions he must sell to get his bonus. Let x = the number of subscriptions Leon sells in the last week of the month.

Step 2

The number of subscriptions per month must be greater than 120. We write 85 + x $latex \ge $ 120

Step 3

We solve the inequality by subtracting 85 from both sides x $latex \ge $ 35 Answer: Leon must sell 35 or more subscriptions in the last week to get his bonus.

Step 4

To check the answer, we see that 85 + 35 = 120. If he sells 35 or more subscriptions the number of subscriptions sold that month will be 120 or more.

Example 6

Virena's Scout Troup 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?

Solution

Step 1

Virena is trying to raise at least $650. Each box of cookies sells for $4.50. Let x = number of boxes sold. The inequality describing this problem is 4.50x $latex \ge $ 650.

Step 2

We solve the inequality by dividing both sides by 4.50; x $latex \ge $ 1444.44 Answer: We round up the answer to 145 since only whole boxes can be sold.

Step 3

If we multiply 145 by $4.50 we obtain $652.50. If Virena's Troop sells more than 145 boxes, they raise more that $650. The answer checks out.

Example 7

The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?

Solution

Step 1

Width = 20 inches; perimeter is at least 180 inches. What is the smallest length that gives that perimeter? Let x = length of the rectangle

Step 2

Formula for perimeter is Perimeter = 2 × length + 2 × width Since the perimeter must be at least 180 inches, we have the following equation. 2 x +2(20)$latex \ge $ 180

Step 3

We solve the inequality. Simplify. 2 x + 40$latex \ge $ 180 Subtract 40 from both sides. 2 x $latex \ge $ 140 Divide both sides by 2. x $latex \ge $e 70 Answer: The length must be at least 70 inches.

Step 4

If the length is at least 70 inches and the width is 20 inches, then the perimeter can be found by using this equation. 2(70) + 2(20) = 180 inches The answer checks out.

Section Summary

  • The general procedure for solving multi-step inequalities is as follows.
    1. Clear parentheses on both sides of the inequality and collect like terms.
    2. Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
    3. Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality if you multiply or divide by a negative number.
  • Inequalities can have multiple solutions, no solutions, or discrete solutions.
ck12, Algebra, Linear Inequalities, " Solving One-Step Inequalities," licensed under a CC BY-NC 3.0 license.