Absolute value inequality

Example 1

Which number could x be?

|x | < 3

Solution

The expression | x | < 3 means all numbers whose distance from zero is less than three. The solution is the interval -3 < x < 3

Example 2

Which number could x be?

|x | > 3

Solution

The expression | x | > 3 means all numbers whose distance from zero is greater than three. The solution is the x < -3 or x > 3

When drawing a solution to a number line, it must be taken into account whether the endpoint, i.e. equality, is included. If equality is not included, as in the examples above, the endpoint is an open dot (circle). When the equality is included, the end point is marked with a completely colored dot.

The above inequality in number line.

Example 3

Solve the equation |x - 3| < 2

Solution

There are two numbers inside the absolute values, x and 3. The above notation means all numbers whose distance from the number three is less than two. Below the solution is marked on a number line, which shows that 1 < x < 5

The equation can also be solved analytically. The expression inside the absolute values |x - 3| < 2 must be greater than -2 or less than 2.

-2 < x - 3 < 2 || + 3

-2 + 3 < x - 3 + 3 < 2 + 3

1 < x < 5

Example 4

Solve the equation |2x + 3| > 1

Solution

The expression inside the absolute values must be greater than 1 or less than -1

2x + 3 > 1 or 2x + 3 < -1

The two inequalities give x > -1 or x < -2

In this case, the inequalities cannot be written together, as the inequalities must remain true from start to finish, as in Example 3.