Absolute value equations and inequalities

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers are at a distance of 3 from 0 on the number line?

3 and −3
Only 3

Key idea

Absolute value is distance.

|x - a|

is the distance between

x

and

a

. So

|x - a| = b

(with

b > 0

) has two solutions,

a - b

and

a + b

;

|x - a| < b

is the interval

a - b < x < a + b

;

|x - a| > b

is the two outer pieces

x < a - b

x > a + b

. If

b < 0

|x - a| = b

has no solution.

Worked example

Solve $|x - 2| = 5$ and $|x - 2| \le 5$ .

x - 2 = 5

x - 2 = -5

x = 7

x = -3

Two cases.

-3 \le x \le 7

Inequality.

Your turn, guided

Your turn: the larger solution of $|x - 4| = 6$ .

10

4 + 6.

Practice ★

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.

How many solutions does the equation |x + 9| = −7 have?
1. None
2. Two
3. One: x = -2

Watch out!

Practice ★★

Solve $|x - (4)| < 4$ . Give the solution set as an interval.
Solve $|x - (9)| < 6$ . Give the solution set as an interval.

Solve $|x - (2)| \geq 2$ . Give the solution set as an interval or a union of intervals.

Exit ticket

Three questions to check you have got it.

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.
Solve $|x - (4)| = 3$ . Enter all solutions.

Summary card

Absolute value equations and inequalities

|x − a| = b: a ± b. Less than: between. Greater than: outside.

Watch out: do not forget the negative case.

End of the lesson

Next lesson: Linear inequalities and systems of inequalities in two variables Back to the chapter

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