Solve Absolute Value Equations | Step-by-Step Calculator

An absolute value equation has the form \( |ax + b| + c = dx + e \). To solve:

Solve both cases, then check each solution in the original equation (extraneous solutions may arise).

📌 The graph shows the left side \( y = |ax+b|+c \) (green) and right side \( y = dx+e \) (blue). Intersection points are solutions.

✧ |ax + b| + c = dx + e ✧

Enter coefficients a, b, c, d, e

Equation: \( |ax + b| + c = dx + e \)

\( a \)

\( b \)

\( c \)

\( d \)

\( e \)

\( |2x + 1| = 2x + 3 \)

⚠️ a cannot be zero (otherwise no absolute value). If a=0, it will be set to 1.

📐 Solution(s)

\( x = -0.5 \)

STEP 1: Split into two cases based on the sign of \( ax + b \)

STEP 2: Solve Case 1 (\( ax + b \ge 0 \)) and check the solution

STEP 3: Solve Case 2 (\( ax + b < 0 \)) and check the solution

STEP 4: Write the solution(s) of the original equation

💡 Graphical Interpretation

The green V-shaped curve is \( y = |ax+b|+c \), the blue line is \( y = dx+e \). Solutions are x-coordinates where the two graphs intersect.

📊 Interactive Graph

y = |ax+b|+c y = dx + e Solution point(s)