Absolute Value Equations
Absolute value equations model distance on the number line. |x โ a| = b means x is exactly b units from a, giving two solutions when b > 0. Master equations, double absolutes, and inequalities with step-by-step solutions.
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|xโa| represents distance from x to a on the number line. |A|=|B| finds points equidistant from both expressions. Always verify solutions by substituting back into the original.
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Why: Absolute value equations appear in temperature tolerances, manufacturing specs, and any problem involving deviation from a target. |T โ 72| โค 3 means temperature stays within 3ยฐF of 72ยฐF.
How: For |ax+b|=c: when c>0, set ax+b=c and ax+b=โc. For |ax+b|=|cx+d|, set the expressions equal or opposite. Inequalities use compound inequalities: |A|โคc โ โcโคAโคc.
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Input Values
Solutions on Number Line
Solution Type
Step-by-Step Solution
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
|xโ3|=5 means x is 5 units from 3, so x=8 or x=โ2.
โ Distance interpretation
|x| is piecewise: x when xโฅ0, โx when x<0.
โ Definition
Key Takeaways
- โข |x| means the distance of x from zero โ it is always non-negative
- โข |ax + b| = c has two solutions when c > 0, one when c = 0, and none when c < 0
- โข |ax + b| = |cx + d| means the two expressions are equidistant from zero โ solve by setting A = B or A = โB
- โข Absolute value equations model distance and deviation in real-world problems (temperature, manufacturing tolerance)
- โข Always verify solutions by substituting back into the original equation
Did You Know?
How It Works
An absolute value equation |ax + b| = c asks: when does the expression ax + b have a distance of c from zero? Since distance is non-negative, c must be โฅ 0 for solutions to exist.
Solving |ax + b| = c
Case 1 (c > 0): Set ax + b = c and ax + b = โc. Solve each linear equation.
Case 2 (c = 0): Set ax + b = 0. One solution.
Case 3 (c < 0): No solution โ |ยท| is never negative.
Solving |ax + b| = |cx + d|
Set ax + b = cx + d or ax + b = โ(cx + d). Solve both linear equations. Check for duplicate or extraneous solutions.
Expert Tips
Use Distance
Think of |x โ a| = b as "x is b units from a." That gives x = a + b or x = a โ b immediately.
Verify Always
Plug each solution back into the original equation. Especially for |A| = |B|, both sides must match.
Isolate First
If you have 2|3x โ 1| + 4 = 10, first get |3x โ 1| = 3, then solve the absolute value equation.
Number Line
Sketch the number line. For |x โ 3| = 5, mark 3 and points 5 units away โ that visual confirms x = 8 and x = โ2.
Reference Table
| Form | Condition | Solutions |
|---|---|---|
| |ax+b| = c | c > 0 | Two: x = (cโb)/a, (โcโb)/a |
| |ax+b| = c | c = 0 | One: x = โb/a |
| |ax+b| = c | c < 0 | None |
| |ax+b| = |cx+d| | โ | Solve A=B and A=โB |
| |ax+b| โค c | c โฅ 0 | Interval [โcโb, cโb]/a |
| |ax+b| โฅ c | c > 0 | Union (โโ, lo] โช [hi, โ) |
FAQ
Why does |x โ 3| = 5 have two solutions?
Because both 8 and โ2 are exactly 5 units from 3 on the number line. Distance is symmetric: |8โ3| = 5 and |โ2โ3| = 5.
When does |ax + b| = c have no solution?
When c < 0. Absolute value is always โฅ 0, so it can never equal a negative number.
How do I solve |x โ 1| = |x + 3|?
Set x โ 1 = x + 3 (gives 0=4, no solution) or x โ 1 = โ(x + 3) โ 2x = โ2 โ x = โ1. The answer is x = โ1.
What is the distance interpretation?
|x โ a| = b means "the distance from x to a is b." So x = a + b or x = a โ b.
Can I have |2x + 1| = 7?
Yes. Set 2x + 1 = 7 โ x = 3, and 2x + 1 = โ7 โ x = โ4. Solutions: x = 3 or x = โ4.
How do inequalities differ from equations?
|ax+b| โค c gives a closed interval (between two points). |ax+b| โฅ c gives a union of two rays (outside the interval).
Quick Reference
Disclaimer: This calculator is for educational purposes. Verify critical results independently. Coefficients should be real numbers; complex solutions are not displayed.
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