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Solve Absolute Value Inequality: |x - 3| ≤ 4
Published on October 7, 2024
Algebra
Solve the absolute value inequality |x - 3| ≤ 4. The solution is -1 ≤ x ≤ 7.
Question
Solve: |x - 3| ≤ 4
Answer
Here’s how to solve the inequality step-by-step:
Understanding Absolute Value
The absolute value of a number is its distance from zero. This means it’s always positive or zero. So, |x| = 3 has two solutions: x = 3 and x = -3.
Solving the Inequality
Because of the absolute value, we need to consider two cases:
Case 1: x - 3 ≥ 0
- If x - 3 is positive or zero, the absolute value doesn’t change anything.
- Our inequality becomes: x - 3 ≤ 4
- Add 3 to both sides: x ≤ 7
Case 2: x - 3 < 0
- If x - 3 is negative, the absolute value makes it positive.
- Our inequality becomes: -(x - 3) ≤ 4
- Distribute the negative sign: -x + 3 ≤ 4
- Subtract 3 from both sides: -x ≤ 1
- Multiply both sides by -1 (and remember to flip the inequality sign!): x ≥ -1
Combining the Solutions
We need to find the values of ‘x’ that satisfy either case 1 or case 2:
- Case 1: x ≤ 7
- Case 2: x ≥ -1
Combining these, the solution is: -1 ≤ x ≤ 7
In LaTeX:
The solution to