Inequalities with absolute value

An inequality with absolute value is about the distance to 0. Remember that the absolute value of a number is always greater than or equal to 0.

 

The definition is:

 

$$ \large |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} $$

 

 

Example 1

We look at the inequality:

 

$$ \large |x| < 3 $$

 

Here we ask: When is \( \large x \) less than 3 units from 0?

The answer is all numbers between -3 and 3:

 

$$ \large -3 < x < 3 $$

 

 

Example 2

Now we look at:

 

$$ \large |x+1| > 2 $$

 

Here it means that the distance from \( \large x \) to -1 must be greater than 2.

This gives two possible intervals:

 

$$ \large x+1 < -2 \quad \text{or} \quad x+1 > 2 $$

 

That is:

 

$$ \large x < -3 \quad \text{or} \quad x > 1 $$

 

 

Example 3

We look at the inequality:

 

$$ \large |2x-4| \leq 6 $$

 

Here the expression must be at most 6 units from 0. It is rewritten as:

 

$$ \large -6 \leq 2x-4 \leq 6 $$

 

We solve step by step:

 

$$ \large -6+4 \leq 2x \leq 6+4 $$

$$ \large -2 \leq 2x \leq 10 $$

$$ \large -1 \leq x \leq 5 $$

 

 

Example 4 (no solution)

Sometimes an absolute value inequality has no solution. For example:

 

$$ \large |x+2| < -1 $$

 

But an absolute value can never be negative, so this inequality has no solution.

 

 

General method

  1. Set up the inequality with absolute value.
  2. Rewrite into a compound inequality (with "<" or "≤") or into two separate inequalities (with ">" or "≥").
  3. Solve the inequalities step by step.
  4. Check the result by inserting test values.
  5. Remember: An absolute value can never be negative. If the inequality requires it, there is no solution.

 

 

Summary

  • Inequalities with absolute value are about distance to 0 or another point.
  • With "<" or "≤" you typically get an interval.
  • With ">" or "≥" you typically get two separate intervals.
  • Sometimes there is no solution.