High School Mathematics 10.2 Solve Absolute Value Inequalities
The equation |x| = 5 means that the distance between x and 0 is 5. The solutions of the equation are 5 and -5 because they are the only numbers whose distance from 0 is 5.
The inequality |x| < 3 means that the distance between x and 0 is less than 3, and
The inequality |x| > 3 means that the distance between x and 0 is greater than 3.
Solving Absolute Value Inequality:
The inequality |ax+b| < c where c > 0 is equivalent to the compound inequality -c < ax+b < c
The inequality |ax+b| > c where c > 0 is equivalent to the compound inequality ax+b < -c
or ax + b > c
In the equalities above, < can be replaced by £ and > can be replaced by ³.
Examples:
Solve |x| ³ 8
The solutions are x ³ 8 and x £ -8
Solve |x| £ 0.7
|x| £ 0.7 -------- original equation
Rewrite as two equations
x £ 0.7 or x ³ -0.7
-0.7£ x £ 0.7
Solve |-4x-5|+3 < 9
|-4x-5|+3 < 9
subtracting 3 both sides of the inequation
|-4x-5|+3 -3 < 9-3
|-4x-5| < 6
-6 < -4x-5 < 6
adding 5 to the inequations
-6+5 < -4x-5+5 < 6+5
-1 < -4x < 11
Dividing by -4 reverses the inequality sign
0.25 > x > -2.75
This can also be written as -2.75 < x < 0.25
Solve |10 - x| > 12
10 - x > 12 or 10 - x < -12
10 -x > 12
subtracting 10 both sides
10 -x-10 > 12-10
-x > 2
Multiplying both sides by (-1) changes the sign
x < -2
10 - x < -12
subtracting 10 both sides
10 -x-10 < -12-10
-x < -22
Multiplying both sides by (-1) changes the sign
x > 22
Therefore x < -2 or x > 22
|x + 2| < -1
Since |x + 2| cannot be negative, |x + 2| cannot be less than -1. So, the solution set is the empty set.
Solution = { }
|2y -1| ³ -4
Since |2y - 1| is always greater than or equal to 0, the solution set is {y|y is a real number}
The graph is the entire number line.
Directions: Solve the following questions. Also write at least 5 examples of your own. Solve them and graph the solution. Click here for additional questions
less than or equal to £
greater than or equal to ³