Multi-Step Inequalities

 

Learning Objective(s)

·         Use the properties of inequality together to isolate variables and solve algebraic inequalities, and express their solutions graphically.

·         Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions.

 

Introduction

Solving multi-step inequalities is very similar to solving equations—what you do to one side you need to do to the other side in order to maintain the “balance” of the inequality. The Properties of Inequality can help you understand how to add, subtract, multiply, or divide within an inequality.

 

Using Properties Together to Solve Inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one step inequalities, the solutions to multi-step inequalities can be graphed on a number line.

 

 

Example

Problem

Solve for p.

4p + 5 < 29

 

 

Begin to isolate the variable by subtracting 5 from both sides of the inequality.

Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.

Answer

 

 

 

To graph this inequality, you draw an open circle at the end point 6 on the number line. The circle is open because the inequality is less than 6 and not equal to 6. The values where p is less than 6 are found all along the number line to the left of 6. Draw a blue line with an arrow on the number line pointing in that direction.

 

 

To check the solution, substitute the end point 6 into the original inequality written as an equation, which is called the related equation, to see if you get a true statement. Then check another solution, such as 0, to see if the inequality is correct.

 

 

Example

Problem

Check that p < 6 is the solution to the inequality 4p + 5 < 29.

 

Check the end point 6 in the related equation.

 

Try another value to check the inequality. Let’s use p = 0.

Answer    p < 6 is the solution to the inequality 4p + 5 < 29.

 

 

Example

Problem

Solve for x.

3x – 7  ≥  41

 

 

Begin to isolate the variable by adding 7 to both sides of the inequality.

Divide both sides of the inequality by 3 to express the variable with a coefficient of 1.

Check

 

First, check the end point 16 in the related equation.

 

 

 

 

 

Then, try another value to check the inequality. Let’s use x = 20.

Answer

 

 

 

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

 

 

Example

Problem

Solve for p.

−6p + 14 < −58

 

Begin to isolate the variable by subtracting 14 from both sides of the inequality.

Divide both sides of the inequality by −6 to express the variable with a coefficient of 1.

Dividing by a negative number results in reversing the inequality sign.

Check

 

 

Check the solution.

First, check the end point 12 in the related equation.

 

 

 

 

Then, try another value to check the inequality. Try 100.

 

Answer

 

 

 

The graph of the inequality p > 12 has an open circle at 12 with an arrow stretching to the right.

 

 

 

Advanced Example

Problem

Solve for x.

 

To isolate the variable, subtract  from both sides of the inequality.

 

 

 

 

 

Then multiply by 3 so that the coefficient in front of the parentheses is 1. Then subtract 3 from both sides.

 

Check

 

Check the solution.

First, check the end point -18 in the related equation.

Now check any value for x that is within the region . We will use .

 

 

 

 

 

 

The statement is true.

Answer

 

 

 

Advanced Question

A student is solving the inequality . If she combines like terms, which of the following inequalities could she see?

 

A)

B)

C)

D)

 

Show/Hide Answer

A)

Incorrect. It looks like you tried to combine like terms but made sign mistakes while adding and subtracting. To isolate the variable term on the right, you need to subtract 7x from both sides; . The correct answer is: .

 

B)

Correct. You correctly combined like terms.  becomes , which is the same as .

 

C)

Incorrect. Remember that you cannot add variable and non-variable terms like you add integers. The correct answer is: .

 

D)

Incorrect. Try subtracting  and 4x from each side before collecting like terms. The correct answer is: .

 

 

 

Using the Distributive Property to Clear Parentheses and Fractions

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

 

 

Example

Problem

Solve for x.

2(3x – 5) ≤ 4x + 6

 

Distribute to clear the parentheses.

Subtract 4x from both sides to get the variable term on one side only.

 

Add 10 to both sides to isolate the variable.

 

Divide both sides by 2 to express the variable with a coefficient of 1.

 

Check

 

   

Check the solution.

First, check the end point 8 in the related equation.

 

 

 

 

 

Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement.

Try 0.

Answer

 

 

 

Example

Problem

Solve for a.

 

Clear the fraction by multiplying both sides of the equation by 6.

 

Add 4 to both sides to isolate the variable.

 

Divide both sides by 2 to express the variable with a coefficient of 1.

Check

 

Check the solution.

First, check the end point 8 in the related equation.

 

 

 

 

 

 

 

 

 

 

Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement.

Try 5.

 

Answer

 

 

 

Advanced Example

Problem

Solve for d.

 

This inequality contains two parentheses. Use the Distributive Property to expand both sides of the inequality.

 

Now that both sides have been expanded, combine like terms and find the range of values for d.

Check

 

Check the solution.

First, check the end point  in the related equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It results in a true statement.

Now try any value for d that is within the region . We will try

 

 

 

 

 

 

This is also a true statement.

Answer

    

 

 

 

Which is the most logical first step for solving for the variable in the inequality:

8x + 7 < 3(2x + 1)

 

A) Reverse the inequality sign.

B) Use the distributive property to clear the parentheses by multiplying each of the terms in the parentheses by 3.

C) Subtract 2x from both sides of the inequality.

D) Divide both sides of the inequality by 3.

 

Show/Hide Answer

A) Reverse the inequality sign.

Incorrect. The inequality sign is reversed as a result of dividing or multiplying both sides of an inequality by a negative number. This usually occurs as a later step in a multi-step inequality and, in the case of this example, will not happen at all. The correct answer is to use the distributive property to clear the parentheses by multiplying each of the terms in the parentheses by 3.

 

B) Use the distributive property to clear the parentheses by multiplying each of the terms in the parentheses by 3.

Correct. The distributive property clears the parentheses so that you can isolate the variable.

 

C) Subtract 2x from both sides of the inequality.

Incorrect. It is necessary to distribute the 3 before subtracting any terms with x in it. The correct answer is to use the distributive property to clear the parentheses by multiplying each of the terms in the parentheses by 3.

 

D) Divide both sides of the inequality by 3.

Incorrect. Dividing both sides by 3, although legal to do, will result in a complicated fraction on the left side of the inequality and is not a logical first step. The correct answer is to use the distributive property to clear the parentheses by multiplying each of the terms in the parentheses by 3.

 

 

Advanced Question

Solve for x.

 

A)

B)

C)

D)

 

Show/Hide Answer

A)

Incorrect. Try substituting any value that is less than or equal to -5 for x into the inequality. You will not get a true statement, so x is not less than or equal to -5. The correct answer is: .

 

B)

Incorrect. You need to evaluate the left side of the inequality and then solve for x. Remember that . The correct answer is: .

 

C)

Incorrect. Make sure you are evaluating the left side of the inequality correctly: . The correct answer is: .

 

D)

Correct. Evaluating the interior parentheses first, you find that . Solving for x, you subtract 25 from both sides and then divide by -10, which requires you to flip the inequality sign from ≤ to ≥.

 

 

 

Summary

Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract either positive or negative numbers to both sides of the inequality.