Absolute Value Equations Calculator

Three boxes, one equation. This absolute value equations calculator solves any equation you can arrange as |a·x + b| = c. That is the shape this absolute value equations calculator is built around. Isolate the absolute value first, then read off the three numbers.

• Coefficient (a) - the number multiplying x inside the bars, such as 2 in $|2x + 3| = 11$.
• Constant (b) - the number added inside the bars, such as 3 in that same equation.
• Right side (c) - the number the absolute value equals, such as 11.

Four rows come back from the absolute value equations calculator - two checks, then two solutions:

• Check 1 tests whether $c \ge 0$. False means the equation has no solution at all.
• Check 2 tests whether $a \ne 0$, so a stray 0 gets caught early.
• Solution 1 comes from the positive branch: $x_1 = (c - b)/a$.
• Solution 2 comes from the negative branch: $x_2 = (-c - b)/a$.

Where do the two branches come from? Removing the bars splits $|a·x + b| = c$ into two plain equations: $a·x + b = c$ and $a·x + b = -c$. The absolute value equations calculator solves both branches in one run.

Try the anchor numbers. Type 2, 3, and 11 to match $|2x + 3| = 11$. The absolute value equations calculator returns $x = 4$ and $x = -7$ instantly. That is the anchor run of this absolute value equations calculator.

Negative right side? Enter $|x + 1| = -3$ as 1, 1, −3 and the first Check row of the absolute value equations calculator reads false: no solution exists.

This page is more than a static absolute value equations calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. “Solve $|2x - 9| = 5$.” The AI fills the three boxes and runs this absolute value equations calculator for you.
• Take a photo of your homework. The AI reads the equation and fills the absolute value equations calculator form.
• Ask about the ideas, not just the number. “Why is there a negative branch?” The AI explains step by step - every number still comes from the calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give one solution and the right side, then ask for the other; this absolute value equations calculator covers both directions. Follow-up questions are always welcome on this page.

Some visitors arrive by typing 'equation solver with absolute value' into a search box. Same page, same three boxes - the absolute value equations calculator does not care how you got here.

Three boxes, four rows, two branches - that is the whole absolute value equations calculator experience.

The absolute value of a number is its distance from zero on the number line. A distance is never negative - that one fact drives every result an absolute value equations calculator reports. Keep the number line in mind and the absolute value equations calculator never surprises you.

So $|x - 5| = 2$ is a distance question in disguise. It asks: which numbers sit exactly 2 units away from 5? The absolute value equations calculator answers in symbols; the number line below answers at a glance - two points qualify. Two points, two branches - the absolute value equations calculator will find both.

Read the picture: 5 sits in the middle, and the two tall red marks flag the answers. $|x - 5| = 2$ has the solutions $x = 3$ and $x = 7$, one on each side of 5. Every absolute value equations calculator output is this picture written in symbols - two answers, symmetric about the center. The absolute value equations calculator just automates the symmetry.

For any algebraic expression $u$ and any positive number $c$: if $|u| = c$, then $u = -c$ or $u = c$. That split is the entire rule the absolute value equations calculator applies. Every absolute value equations calculator row traces back to that one split.

$|a x + b| = c \quad\Longleftrightarrow\quad a x + b = c \;\text{ or }\; a x + b = -c$

Solving the two branches gives the pair of formulas the absolute value equations calculator prints as its last two rows:

$x_1 = \frac{c - b}{a} \qquad\qquad x_2 = \frac{-c - b}{a}$

Every symbol matches a box in the absolute value equations calculator above - one letter, one box.

The manual routine has four steps, and the absolute value equations calculator walks the last three with you:

1. Isolate the absolute value expression - get it alone on one side.
2. Write the two equivalent equations, one with $+c$ and one with $-c$.
3. Solve each equation - one linear step apiece.
4. Check each solution in the original equation - the absolute value equations calculator builds its two Check rows on this habit.

Four steps, two answers - the absolute value equations calculator runs them the moment you click.

Solve $|2x + 3| = 11$ with the absolute value equations calculator, then by hand to see both branches at work.

1. Match the form: $a = 2$, $b = 3$, $c = 11$.
2. Positive branch: $2x + 3 = 11$, so $2x = 8$ and $x = 4$.
3. Negative branch: $2x + 3 = -11$, so $2x = -14$ and $x = -7$.
4. Check: $|2 \cdot 4 + 3| = |11| = 11$ and $|2 \cdot (-7) + 3| = |-11| = 11$.

Both solutions verify. Type 2, 3, 11 into the absolute value equations calculator above; the absolute value equations calculator returns the same pair instantly.

Now the picture above becomes an equation to solve: $|x - 5| = 2$.

1. Match the form: $a = 1$, $b = -5$, $c = 2$.
2. Positive branch: $x - 5 = 2$, so $x = 7$.
3. Negative branch: $x - 5 = -2$, so $x = 3$.
4. Check: $|7 - 5| = 2$ and $|3 - 5| = |-2| = 2$.

The two answers sit 2 units from 5, one on each side - the number line predicted both. Enter 1, −5, 2 in the absolute value equations calculator and the absolute value equations calculator returns the same pair.

One more read: 7 is 2 above 5, and 3 is 2 below it. The branches are just the two directions away from the center, and the absolute value equations calculator checks both.

Solve $|x + 1| = -3$.

1. Match the form: $a = 1$, $b = 1$, $c = -3$.
2. Read the first Check row before anything else: $c \ge 0$ is false.
3. An absolute value is never negative, so no x can make the equation true.

No branches, no arithmetic - the answer is no solution. The rule is blunt: an absolute value cannot be negative. The absolute value equations calculator shows the same verdict through its Check row. The absolute value equations calculator will not fake a solution. The two solution rows still print numbers - discard them when the Check is false.

Three slips the absolute value equations calculator catches every day:

1. Keeping only the positive branch. An equation like $|u| = c$ with $c > 0$ always splits into two. Drop the negative branch and you lose half the answers - the absolute value equations calculator always reports both.
2. Splitting before isolating. Take $|2x - 3| - 4 = 3$. Add 4 to both sides first, so the equation becomes $|2x - 3| = 7$; only then split. The absolute value equations calculator expects the isolated shape too. Split too early and the stray $-4$ corrupts both branches.
3. Missing the no-solution case. If the isolated right side is negative, stop - there is nothing to solve for. Writing $x = -3$ as a solution of $|x + 1| = -3$ is the classic slip the Check row of the absolute value equations calculator exists to catch.

Avoid these three and the absolute value equations calculator becomes a checking tool, not a crutch.