Absolute Value Inequalities Calculator

Fill four boxes, click Calculate. This absolute value inequalities calculator solves the standard form $|ax + b| \;\square\; c$, and the dropdown picks the sign in the middle. Fill the boxes as the letters appear in your problem and press Calculate once. One pass of the absolute value inequalities calculator settles the question - no case sorting, no second run.

• Coefficient of x (a) - the number multiplying $x$ inside the bars, such as 2 in $|2x + 1| \le 9$; a negative $a$ goes in as-is - the absolute value inequalities calculator keeps your sign
• Constant (b) - the number added to $ax$, such as 1
• Right-hand side (c) - the number on the far side of the sign, such as 9, which the absolute value inequalities calculator compares against $ax + b$
• Comparison sign - choose $<$, $>$, $\le$ or $\ge$; the absolute value inequalities calculator uses it to shape the answer

Three rows come back. Endpoint 1 solves $ax + b = c$ and Endpoint 2 solves $ax + b = -c$. Together they are the two edges of the answer, and the absolute value inequalities calculator prints both at once, matched to the $+c$ and $-c$ branches of the table below. The band width row prints $2c/|a|$, the length of the stretch, so you can check the endpoints against it at a glance. If the two endpoint rows ever come out equal, the absolute value inequalities calculator is telling you the answer has collapsed to a single point - a band width of zero.

Reading the rows is where the absolute value inequalities calculator saves you the case sorting. For a less-than problem the answer is the stretch between the two endpoints. For a greater-than problem the answer is two rays running outward. The absolute value inequalities calculator still finds the same two edges - only the joining word changes, from and to or.

This page is more than a static absolute value inequalities calculator - it is also an AI tutor: the question box takes plain words, homework photos, and follow-ups. Ask it to fill the four boxes from a word problem. The absolute value inequalities calculator then runs on exactly the numbers the AI typed in, so audit each field before trusting the result. Ask it to explain any step in words. Every number still comes from the form on this page.

One habit makes the absolute value inequalities calculator twice as useful: read $c$ before you press Calculate. A negative $c$ changes the whole story - read it first, and the absolute value inequalities calculator's rows will always mean what you expect. The sections below show exactly how.

Read the bars as a distance. $|2x + 1| \le 9$ asks: which $x$ keep the value $2x + 1$ within 9 steps of zero? Each qualifying $x$ sits in one connected stretch of the number line. An absolute value inequalities calculator exists to pin down the two ends of that stretch, and this absolute value inequalities calculator does it from three numbers and a sign. The absolute value inequalities calculator turns that single question into two boundary equations, and nothing else.

The picture shows the answer to $|2x + 1| \le 9$: a closed dot at $-5$, a closed dot at $4$, and every point between them shaded. The absolute value inequalities calculator confirms the same pair from $a = 2$, $b = 1$, $c = 9$. Asked which $x$ qualify, the answer is exactly the shaded band. Reading the drawing beside the rows is the absolute value inequalities calculator showing the same answer twice.

Distance also explains why the answer never falls into three or more pieces. A distance either fits inside its bound or it does not, so one run of the absolute value inequalities calculator sorts every point on the line into in or out. That is why a single run of the absolute value inequalities calculator is always enough.

One table runs the whole topic. For an expression $u$ inside the bars and a positive bound $c$, the rewrite is fixed - and the absolute value inequalities calculator applies it before anything else. This table is also the wiring inside the absolute value inequalities calculator.

Less-than signs keep the answer between the two edges: one stretch, joined by and. Greater-than signs push it outside: two rays, joined by or. Every absolute value inequalities calculator is an application of this split. The table also explains the two endpoint rows - they are the $-c$ edge and the $+c$ edge that the absolute value inequalities calculator solves for.

The letters on this page match the form fields exactly, so a problem can be copied straight into the absolute value inequalities calculator:

The endpoints solve the two boundary equations at once. Where $ax + b = c$, $x = \dfrac{c - b}{a}$. Where $ax + b = -c$, $x = \dfrac{-c - b}{a}$. Those two values are the first two rows the absolute value inequalities calculator returns, in that order - one absolute value inequalities calculator pass, both boundaries solved. The third row is the band width $\dfrac{2c}{|a|}$ - the gap between the two endpoints, taken as the absolute value of their difference.

Which endpoint lands on the left depends on the sign of $a$. With $a = 2$, $b = 1$, $c = 9$ the rows come out as $4$ and $-5$, so $-5$ is the left edge. Change to $a = -2$ and the absolute value inequalities calculator hands back the same pair in mirrored order. Sorting them by hand is the one step the absolute value inequalities calculator leaves to you.

The workflow stays short either way. Sort the two endpoint values, then let the and/or table decide what the absolute value inequalities calculator should draw between them and beyond them.

Three kinds of input deserve a second look before you trust any output of the absolute value inequalities calculator.

• Negative $c$ with $<$ or $\le$. A distance is never negative, so $|2x - 1| < -4$ has no solution. The endpoint rows still compute, but nothing satisfies the inequality - the absolute value inequalities calculator cannot read $c$ for you.
• Negative $c$ with $>$ or $\ge$. The reverse: $|x + 3| > -2$ is true for every real $x$, because a distance is always zero or more - the endpoint rows still compute, and the or-rule then lets every real $x$ qualify.
• $a = 0$. The bars then hold a bare constant such as $|5| < 9$, and there is no $x$ left to solve for. Keep $a$ nonzero so the absolute value inequalities calculator sees a real expression in $x$.

Solve $|2x + 1| \le 9$, then write the answer three ways.

1. Split: $-9 \le 2x + 1 \le 9$ - a less-than case, so one stretch joined by and.
2. Subtract 1 from all three parts: $-10 \le 2x \le 8$.
3. Divide by 2: $-5 \le x \le 4$ - the same pair the absolute value inequalities calculator prints. In interval notation: $[-5, 4]$.

Now let the absolute value inequalities calculator confirm it with $a = 2$, $b = 1$, $c = 9$. Endpoint 1 reads $\dfrac{9 - 1}{2} = 4$ and Endpoint 2 reads $\dfrac{-9 - 1}{2} = -5$ - the same pair, delivered without any case sorting. The band width row prints $\dfrac{2 \cdot 9}{2} = 9$, and indeed $4 - (-5) = 9$. Spot-check an interior point: $x = 0$ gives $|1| = 1 \le 9$, true as promised. The figure above also shows what $\le$ buys you: both endpoints are solid dots, which is exactly how the absolute value inequalities calculator marks an edge that counts.

Solve $|3x + 6| \ge 12$, then describe the solution set.

1. Split into two branches: $3x + 6 \ge 12$ or $3x + 6 \le -12$.
2. First branch: $3x \ge 6$, so $x \ge 2$.
3. Second branch: $3x \le -18$, so $x \le -6$. Answer: $x \le -6$ or $x \ge 2$.

Set $a = 3$, $b = 6$, $c = 12$ in the absolute value inequalities calculator. Endpoint 1 is $\dfrac{12 - 6}{3} = 2$ and Endpoint 2 is $\dfrac{-12 - 6}{3} = -6$, so the picture below needs no editing: closed dots at $-6$ and $2$, shading outward on both sides. The absolute value inequalities calculator found the same two edges your algebra did. Run it again with $c = 11$ and the right edge slides from $2$ to $\dfrac{5}{3}$ - the absolute value inequalities calculator tracks the bound as $c$ moves.

Only the sign changed from Example 1 - $\le$ became $\ge$ - and the answer moved from one band to two rays. That is exactly the difference the absolute value inequalities calculator exists to get right. Ask it for both cases and compare the two drawings.

Each of these slips survives sloppy algebra but not a careful pass through the absolute value inequalities calculator.

1. Solving only the positive branch. From $|3x + 6| \ge 12$, keeping just $x \ge 2$ loses half the answer. Every $x \le -6$ works too - test $x = -7$: $|-15| = 15 \ge 12$. The two endpoint rows always come as a pair, so the absolute value inequalities calculator makes the loss visible.
2. Writing the band backwards. From $-9 \le 2x + 1$ some students continue with $9 \ge 2x + 1 > -9$ and end up chaining the signs in mixed directions. Keep one direction per line, or let the two endpoint rows of the absolute value inequalities calculator keep the order straight for you.
3. Ignoring a negative right-hand side. Announcing that $|2x - 1| < -4$ solves to some pair of endpoints forgets that the bars never produce a negative. Check $c$ first: negative with $<$ means no solution, negative with $>$ means everything - the absolute value inequalities calculator will print rows either way.

The absolute value inequalities calculator returns its endpoint rows side by side, so mistakes 1 and 2 surface the moment you compare them. Mistake 3 is the one check no absolute value inequalities calculator can do for you - read the sign of $c$ before you press Calculate.