Solving Absolute Value Inequalities Part 1Honors Math – Grade 8

Absolute Value Inequalities with <

When solving an inequality of the form |x|<n, consider the following cases:

1. The expression inside the absolute value symbols is positive.

2. The expression inside the absolute value symbols is negative.

5xThis means the distance from zero is less than 5 units.

5 units 5 units

This shows an intersection. Write an inequality for each situation. x > -5 x < 5KEY CONCEPT Absolute Value Inequalities with <

ax + b > -candax + b < cmeans|ax + b|<c

ax + b > -candax + b < cmeans|ax + b|<c

Solve the open sentence. Then graph the solution set. 45 g

Write the inequality as a compound inequality using “and.”

g + 5 < 4 g + 5 > -4

Solve each inequality.

-5 -5 -5 -5

g < -1 g > -9

Therefore, g < -1 and g > -9.

Graph the solution set.The solution set represents an intersection.

The solution set is: -9 < g < -1

Solve the open sentence. Then graph the solution set. 28 n

Write the inequality as a compound inequality using “and.”

n – 8 < 2 n - 8 > -2

Solve each inequality.

+8 +8 +8 +8

n < 10 n > 6

Therefore, n < 10 and n > 6

Graph the solution set.The solution set represents an intersection.

The solution set is: 6 < n < 10

Solve the open sentence. Then graph the solution set. 352 c

Write the inequality as a compound inequality using “and.”

2c + 5 < 3 2c + 5 > -3

Solve each inequality.

-5 -5 -5 -5

2c < -2 c < -1

2c > -8 c > -4

Therefore, c < -1 and c > -4.

Graph the solution set.The solution set represents an intersection.

The solution set is: -4 < g < -1

75 cSolve

Recall that the absolute value of a number is the distance from zero. This means that the absolute value of a number is always positive!

Since l x + 2 l cannot be negative, l x + 2 l cannot be less than -7. So the solution is

This is the symbol for the empty set.

It means there is NO SOLUTION!