Absolute Value Equation Solver

About Absolute Value Equation Solver

To solve |f(x)| = c, solve both f(x) = c and f(x) = -c. If c < 0, there is no solution.

How to use this calculator

1. Enter the coefficient of x, the constant term, and the right side value c.
2. Check that the equation is in the form |ax + b| = c.
3. Read the two possible cases, ax + b = c and ax + b = -c.
4. Solve each linear equation and combine the solutions, or report no solution if c < 0.

The formula explained

The rule |ax + b| = c Rightarrow ax + b = c means the expression inside the absolute value can equal either positive c or negative c. This turns one absolute value equation into two simpler linear equations.

• a = the coefficient of x
• b = the constant inside the absolute value
• c = the nonnegative number on the right side
• x = the variable you are solving for

Step by step method

1. First, check whether c < 0. If it is, there is no solution because an absolute value cannot be negative.
2. Write two equations: ax + b = c and ax + b = -c.
3. Solve each equation for x by isolating the variable.
4. Check your answers in the original equation if you want to confirm them.

Worked example

Problem. Solve |2x - 5| = 7.

1. Set up the two cases: 2x - 5 = 7 and 2x - 5 = -7.
2. Solve the first equation, 2x = 12, so x = 6. Solve the second equation, 2x = -2, so x = -1.
3. Both values work, so the solution set is {x = 6, x = -1}.

Tips and common mistakes

• If the right side is negative, like |2x - 5| = -3, the equation has no solution.
• Be careful with the minus sign in the second case, because ax + b = -c changes the sign of every term on the right side.

Frequently asked questions