How To Solve Inequalities Involving Absolute Values?

By Robert O

Handling inequalities that involve absolute values follow the same steps for solving equations with absolute values. You can refer to our previous tutorial as a prerequisite to this lesson. However, we will again list all the steps here and use them in solving some examples.

Inequalities simply refer to the use of signs (≤, ≥, <, >) instead of an equal sign (=) to write an equation. The only difference that comes with this representation is that we cannot have one solution but a range of values that satisfy the inequality.

How to solve absolute value problems with inequalities?

The following are the steps that you need to follow to solve absolute value problems:

Step 1: Taking the absolute value term to one side of the equation.

Step 2: Checking if the value on the other side is negative or positive. If it is negative, then there is no solution and no further action. If it is positive, proceed with the operation as in step 3.

Step 3: Removal of the absolute bars and setting a compound inequality equation. This is an equation with two inequality symbols.

The setting up of a compound inequality depends on the type of inequality sign in the original equation. We will see how that works using some examples.

How to solve inequality equations involving greater than and less than (<) or equal to ( ≥)?

Example 1:

Solve |x + 2| – 4 < 6.

Solution

We start by taking the absolute value term to one side of the equation: |x + 2| < 10.

We then check if the number on the other side of the equation is negative. 10 is a positive value, meaning the inequality has a solution.

We use this general rule in setting up a compound equation:

If |a| < b, then -b < a < b, given b > 0.

If |a| ≤ b, then -b ≤ a ≤ b, given b > 0.

So, we can write a compound inequality from |x + 2| < 10 as -10 < x + 2 < 10

Solving the compound inequality for x by subtracting 2 from both sides gives:

-12 < x < 8

All values between -12 and +8 satisfy the equation.

Example 2:

Solve |7 x + 2| ≤ 1

Solution

We go straight to step 2, where we find that the number on the other side of +2. We proceed with the operation since inequality has a solution.

Drop the absolute value bars and write a compound inequality using the formula If |a| ≤ b, then -b ≤ a ≤ b.

Solve for x by subtracting 2 from both sides and dividing by 7.

The interval notation is

How to solve inequality equations involving greater than and greater than (>) or equal to (≥)

What changes here is how we write the compound inequality equation. Instead, we write two separate equations. We use this general formula:

If |a| > b, then a > -b or a > b, given b > 0

If |a| (≥) b, then a ≥ -b or a ≥ b, given b > 0

Example 3:

Solve |2 x – 2| > 6

Solution

The value on the other side is greater than 0. We use the formula or general rule of thumb to write two different inequalities:

2 x – 2 > 6     or    2 x – 2 < – 6

2 x > 8          or     2 x < – 4

x > 4             or      x > – 2

Example 4:

Solve |4 x + 8| ≥ 1

Solution

We can quickly see that there are solutions to the equation. So, we will write our two inequalities straightaway.

4 x + 8 ≥ 1       or       4 x + 8 ≤ – 1   

4x ≥ – 7            or       4 x ≤ – 9

x ≥ – 7/4          or       x ≤ – 9/4

From the solution, values of x range from negative infinity to -9/4. The other solution also suggests that x values range from -7/4 to positive infinity. The correct notation for the two statements is:

Example 5: 

Solve |5 x + 6| + 4 < 1

Solution

Start by isolating the absolute value on one side. The result is |5 x + 6| < -3

Using step 2, we see that the equation has no solution since the number on the other side is negative. But what happens should we choose to proceed with the operation? Let’s find out!

The values to the left will always yield a positive number. The one on the right is always negative.

Positive< negative. A false statement like this one confirms that the equation has no solution.

Remarks

By following the steps and using the formula, solving inequalities involving absolute values can never be a problem. Regular practice is the only way to master the concept and save you more time solving such problems.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.