Absolute Value Calculator

What Is Absolute Value?

Absolute value is the distance of a number from zero on the number line. It is written with vertical bars: |−7| = 7 because −7 sits 7 units from zero, and |5| = 5. Since distance is never negative, an absolute value is always zero or positive — never negative.

• |-7| = 7 — the minus sign disappears, the size remains
• |3 − 9| = |−6| = 6 — evaluate inside first, then take the size
• |0| = 0 — zero is the only number whose absolute value is zero

How to Solve Absolute Value Equations

To solve an equation like 3|2x+1|+4=25, follow four steps. The calculator above does all of this for any equation of the form a|bx+c|+d=e and shows each step.

1. Isolate the bars. Subtract 4 from both sides, then divide by 3: 3|2x+1| = 21, so |2x+1| = 7.
2. Split into two equations. Drop the bars twice — once positive, once negative: 2x+1 = 7 or 2x+1 = −7.
3. Solve each one. From 2x+1 = 7 you get x = 3; from 2x+1 = −7 you get x = −4.
4. Check both answers. 3|2(3)+1|+4 = 3|7|+4 = 25 ✓, and 3|2(−4)+1|+4 = 3|−7|+4 = 25 ✓. Both work.

Absolute Value Equations vs Inequalities

Equations ask “which points?”, inequalities ask “which range?”. Solving |x−2| < 3 gives every number within 3 units of 2, while flipping the sign gives everything outside that range.

Common Mistakes to Avoid

• Forgetting the negative case. |2x+1| = 7 needs both 2x+1 = 7 and 2x+1 = −7. Solving only the positive half loses x = −4.
• Splitting before isolating. In 3|2x+1|+4=25 you must reach |2x+1| = 7 first. Splitting while the 3 and +4 are still attached gives wrong answers.
• Confusing “no solution” with “zero”. |x| = 0 has exactly one solution (x = 0), but |x| = −1 has none — bars can never equal a negative number.

Where Absolute Value Shows Up

• Distance: |a − b| is the gap between two numbers, on maps and timelines alike.
• Tolerance: a part labeled 10 ± 0.5 mm passes when |measured − 10| ≤ 0.5.
• Error bounds: forecasts report |prediction − actual| as the size of the miss.