Section 1.7
Absolute Value Inequalities
“LESS THAN” or WITHIN
Find all values of x that are less than (or within) 5 units away from 0.
Solve the inequality and express in interval notation.
a.
x 5
b.
x 5
c.
|𝑥 | < −5
Steps for Solving “Less than” Absolute Value Inequalities
1) Rewrite the equation to be: |𝒊𝒏𝒔𝒊𝒅𝒆| < 𝒄.
2) a) If c is positive, set up compound inequality −𝒄 < 𝒊𝒏𝒔𝒊𝒅𝒆 < 𝒄. Solve the inequality and express
answer in interval notation.
b) If c is negative, then there is NO SOLUTION.
NOTE: If |𝒊𝒏𝒔𝒊𝒅𝒆| ≤ 𝒄, then set up as −𝒄 ≤ 𝒊𝒏𝒔𝒊𝒅𝒆 ≤ 𝒄.
Example 1: Solve the inequalities and express in interval notation.
a)
|𝑥 − 2| + 1 < 5
“GREATER THAN”
b)
Find all values of x that are more than (or = to) 5 units away from 0.
Solve the inequalities and express in interval notation.
a.
b.
x 5
x 5
c. |𝑥 | > −5
|𝑥 − 7| + 5 < 1
Steps for Solving “Greater than” Absolute Value Inequalities
1) Rewrite the equation to be: |𝒊𝒏𝒔𝒊𝒅𝒆| > 𝒄.
2)
a) If c is positive, set up two inequalities 𝒊𝒏𝒔𝒊𝒅𝒆 > 𝒄 OR 𝒊𝒏𝒔𝒊𝒅𝒆 < −𝑐.
Solve the inequalities and express answer in interval notation. Use the ∪ symbol to unite.
b) If c is negative, then the solution is ALL REAL NUMBERS.
NOTE: If |𝒊𝒏𝒔𝒊𝒅𝒆| ≥ 𝒄, then set up the inequalities as 𝒊𝒏𝒔𝒊𝒅𝒆 ≥ 𝒄 OR 𝒊𝒏𝒔𝒊𝒅𝒆 ≤ −𝒄.
Example 2:
a)
Solve the inequalities and express in interval notation.
|𝑥 + 3| − 1 ≥ 8
If you are given:
|𝒊𝒏𝒔𝒊𝒅𝒆| > 𝑐.
|𝒊𝒏𝒔𝒊𝒅𝒆| ≥ 𝑐.
|𝒊𝒏𝒔𝒊𝒅𝒆| < 𝑐.
|𝒊𝒏𝒔𝒊𝒅𝒆| ≤ 𝑐.
b)
|𝑥 − 7| + 5 > 1
Use this setup:
𝒊𝒏𝒔𝒊𝒅𝒆 > 𝑐 OR 𝒊𝒏𝒔𝒊𝒅𝒆 < −𝑐.
𝒊𝒏𝒔𝒊𝒅𝒆 ≥ 𝑐 OR 𝒊𝒏𝒔𝒊𝒅𝒆 ≤ −𝑐.
−𝒄 < 𝒊𝒏𝒔𝒊𝒅𝒆 < 𝒄
−𝒄 ≤ 𝒊𝒏𝒔𝒊𝒅𝒆 ≤ 𝒄.
Example 3: Solve the inequalities and express in interval notation.
a) |2𝑥 − 1| ≤ 7
c)
|2𝑎 + 1| + 4 ≥ 7
b)
d)
|1 + 4𝑥| > 7
|6𝑦 − 1| − 4 < 2