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1. Equations & Inequalities in One Variable
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United States of AmericaFL
Grade 912

Honors: 1.08 Absolute value inequalities

Lesson
Practice
Lesson

Concept summary

The absolute value of a number is a measure of the size of a number, and is equal to its distance from zero (0), which is always a non-negative value. Absolute value is sometimes called "magnitude".

An absolute value inequality is an inequality containing the absolute value of one more variable expressions.

Inequality

A mathematical relation that compares two non-equivalent expressions

Solutions to absolute value inequalities usually involve multiple inequalities joined by one of the keywords "and" or "or". Solutions with two overlapping regions joined by "and" can be rewritten as a single compound inequality:

\left|x\right| \geq 1 has solutions of x \leq -1 \text{ or } x \geq 1 or in interval notation \left(-\infty, -1\right] \cup \left[-1,\infty\right).

\left|x\right| < 3 has solutions of x > -3 \text{ and } x < 3 which is equivalent to -3 < x < 3 or in interval notation \left(-3,3\right).

In general, for an algebraic expression p(x) and k>0, we have:

  • \left|p(x)\right|<k can be written as -k<p(x)<k
  • \left|p(x)\right|\leq k can be written as -k\leq p(x) \leq k
  • \left|p(x)\right|> k can be written as p(x)<-k or p(x)> k
  • \left|p(x)\right|\geq k can be written as p(x)\leq-k or p(x)\geq k

Solutions to absolute value inequalities can also be represented graphically using number lines:

-4-3-2-101234
\left|x\right| \geq 1
-4-3-2-101234
\left|x\right| < 3

Worked examples

Example 1

Consider the inequality \left|x\right| > 2:

a

Represent the inequality \left|x\right| > 2 on a number line.

Approach

This inequality represents values of x which are "more than 2 units away from 0". To plot this, we will need to use two regions. Also note that this inequality doesn't include the endpoints, so we will use unfilled points to show this.

Solution

-5-4-3-2-1012345

Reflection

We can give a quick check of the answer by thinking about whether this is an "and"-type inequality or an "or"-type inequality.

In this case, the inequality \left|x\right| > 2 has a solution of x < -2 \text{ or } x > 2. Our solution on the numberline has two distinct parts, which matches what we expect for an "or"-type inequality.

b

Rewrite the solution to \left|x\right| > 2 in interval notation.

Approach

We can use either the number line or x < -2 \text{ or } x > 2 to help write this in interval notation.

Solution

The x < -2 part can be written as \left(-\infty, -2\right).

The x >2 part can be written as \left(-2,-\infty\right).

Since this is an "or", not "and", we will find the union of these two sets to give: \left(-\infty, -2\right) \cup \left(-2,-\infty\right)

Example 2

Consider the inequality \left|4x - 5\right| \leq 3.

a

Solve the inequality for x. Express your solution using interval notation.

Approach

In order to solve this inequality, it will be easier to first remove the absolute value by rewriting the inequality as a compound inequality. We can then solve as normal.

Solution

\displaystyle \left|4x - 5\right|\displaystyle \leq\displaystyle 3
\displaystyle -3 \leq 4x - 5\displaystyle \leq\displaystyle 3Rewrite as a compound inequality
\displaystyle 2 \leq 4x\displaystyle \leq\displaystyle 8Add 5 to each part
\displaystyle \frac{1}{2} \leq x\displaystyle \leq\displaystyle 2Divide throughout by 4

So the solutions are "all values of x between \dfrac{1}{2} and 2 inclusive". We can express this using interval notation as the interval \left[\frac{1}{2},\, 2\right]

b

Represent the solution set on a number line.

Approach

The solution set for this inequality consists of a single interval, so it will only need one region on the numberline. The endpoints are also included this time, which we represent using filled points.

Solution

-4-3-2-101234

Reflection

We can give a quick check of the answer by thinking about whether this is an "and"-type inequality or an "or"-type inequality.

In this case, we found the solution to the inequality \left|4x - 5\right| \leq 3 to be the compound inequality \dfrac{1}{2} \leq x \leq 2. A compound inequality is a way of writing an "and"-type inequality, which matches the fact that the numberline has one region between two endpoints.

Example 3

Write an absolute value inequality to represent the set of "all real numbers x which are at least 5 units away from 12".

Approach

We want to break down the wording of the set into its key parts. These key parts are:

  • at least
  • away from 12
  • 5 units away

Each of these key parts tells us information about the form of the inequality.

Solution

Let's interpret these key parts:

  • "At least" is another way of saying "greater than or equal to", which tells us the inequality symbol that we want to use.
  • "Away from 12" means that we want to compare the distance between x and 12. We can represent this as \left|x - 12\right|.
  • "5 units away from" means that the absolute value will be compared to 5.

Putting this all together, we get "the distance between x and 12 is greater than or equal to 5", which we can represent as the absolute value inequality \left|x - 12\right| \geq 5

Outcomes

MA.912.AR.2.6

Given a mathematical or real-world context, write and solve one-variable linear inequalities, including compound inequalities. Represent solutions algebraically or graphically.

MA.912.AR.4.1

Given a mathematical or real-world context, write and solve one-variable absolute value equations.

MA.912.AR.4.2

Given a mathematical or real-world context, write and solve one-variable absolute value inequalities. Represent solutions algebraically or graphically.

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