Absolute Value Calculator
The absolute value calculator isolates the bars, splits the problem into its two cases, and solves each one exactly. It also gives the absolute value of a single number.
Solution
- Step 1: isolate the absolute value Subtract 4 from both sides, then divide both sides by 2.
- 2|x - 3| + 4 = 10, then 2|x - 3| = 6, then |x - 3| = 3
- Step 2: the positive case What is inside the bars equals 3.
- x - 3 = 3, so x = 6
- Step 3: the negative case What is inside the bars equals -3. Both have an absolute value of 3.
- x - 3 = -3, so x = 0
- Check Each answer substituted back into the original equation.
- x = 0: 10 = 10; x = 6: 10 = 10
- Graph The solutions are where the graph meets the horizontal line y = 10. A V crosses a horizontal line twice, touches it once at the corner, or misses it.
- y = 2|x - 3| + 4 is a V with its corner at (3, 4)
- Every value is an exact fraction. Decimals are read as written, so 0.1 is exactly 1/10.
How the Absolute Value Calculator works
The absolute value of a number is its distance from zero, so it is never negative: |7| = 7, |-7| = 7 and |0| = 0. That one fact drives every method below.
Solving an absolute value equation. Take 2|x - 3| + 4 = 10, the default here.
1. Isolate the absolute value. Subtract 4 from both sides to get 2|x - 3| = 6, then divide by 2 to get |x - 3| = 3.
2. Check the right-hand side. If it were negative there would be no solution at all, because no absolute value equals a negative number. If it were 0 there would be exactly one. Here it is 3.
3. Split into two cases, because two numbers have an absolute value of 3: x - 3 = 3, which gives x = 6, and x - 3 = -3, which gives x = 0.
4. Check both in the original equation: 2|6 - 3| + 4 = 10 and 2|0 - 3| + 4 = 10.
The most common mistake is splitting before isolating. Writing 2(x - 3) + 4 = 10 and 2(x - 3) + 4 = -10 gives x = 6 and x = -4, and -4 does not satisfy the original equation. Isolate first, every time.
Solving an absolute value inequality. After isolating, the shape of the answer depends on the direction:
|u| < R means u is within R of zero, which is one compound inequality, -R < u < R, and the answer is a single interval.
|u| > R means u is more than R from zero in either direction, which is two inequalities joined by "or", u < -R or u > R, and the answer is two separate rays.
The same goes for less than or equal and greater than or equal, with the endpoints included. Dividing by a negative number at any step reverses the inequality. When R is negative, the "less than" forms have no solution and the "greater than" forms hold for every x.
As a solving absolute value equation calculator it shows each of those steps for the equation you enter, and as an inequality solver it does the same for <, ≤, > and ≥.
Exact answers. Every coefficient is read as an exact fraction, so |3x + 1| = 2 gives x = 1/3 and x = -1, not 0.3333333. Decimals are read as written.
On a graph, y = p|ax + b| + k is a V with its corner at x = -b/a. The solutions of the equation are where the V crosses the horizontal line y = c, which is why there can be two, one or none.
Absolute Value Calculator FAQ
How do you solve an absolute value equation?
Isolate the absolute value, then split into two equations: the inside equals the right-hand side, and the inside equals its negative. Solve both and check them. For |x - 3| = 3, x - 3 = 3 gives 6 and x - 3 = -3 gives 0.
When does an absolute value equation have no solution?
When, after isolating, the absolute value equals a negative number. |2x + 1| = -5 has no solution because an absolute value is a distance and is never negative.
How do you solve an absolute value inequality?
Isolate the absolute value. For |u| < R write -R < u < R and solve the compound inequality. For |u| > R write u < -R or u > R and solve each. Reverse the inequality whenever you multiply or divide by a negative number.
What is the absolute value of a negative number?
The same number without the minus sign. |-7.5| = 7.5 and |-3/4| = 3/4. It is the number's distance from zero on the number line.
Why do absolute value equations have two solutions?
Because two numbers lie at any positive distance from zero, one on each side. |u| = 3 means u = 3 or u = -3, and each gives its own value of x.