Type a linear, compound, quadratic, rational or absolute value inequality to solve it step by step, with the answer in inequality and interval notation and on a number line. It flips the sign when dividing by a negative (3x + 2 > 5x − 4), uses a sign chart for quadratic and rational inequalities like x² − 4 < 0, and splits absolute values: |2x − 1| < 5 becomes −5 < 2x − 1 < 5.
The one rule that differs from equations
Solve an inequality like an equation, except: multiplying or dividing both sides by a negative number reverses the sign. From -2x \ge 4 you get x \le -2. The calculator avoids the flip where it can by gathering x on the side where its coefficient is positive, and explains it whenever it happens.
Example: -3x + 5 > 11
-3x > 6 \quad\Rightarrow\quad x < -2 \quad (\text{divided by } -3\text{, sign reversed})Test one number from the answer to be sure: x = −3 gives 9 + 5 = 14, which is more than 11. A number outside it, x = 0, gives 5 – not more than 11. Open it in the calculator.
“And” and “or”: compound inequalities
A double inequality is an “and” – both parts must hold. Do the same thing to all three parts: from -1 < 2x + 3 \le 9, subtract 3 and halve to get -2 < x \le 3, the interval (−2, 3]. An “or” needs only one part to hold, so the answers are joined: 2x − 1 > 5 or x + 4 < 2 gives x < −2 or x > 3, two separate rays. When the two parts of an “or” overlap to cover everything – x > 1 or x < 4 – every number works.
Inequalities in word problems
“At most”, “no more than” and “maximum” mean ≤; “at least”, “no less than” and “minimum” mean ≥. A data plan costs ₹200 a month plus ₹15 for each extra GB, and you can spend at most ₹500: 200 + 15x ≤ 500 gives x ≤ 20, so up to 20 extra GB. When the answer must be a whole number of things – tickets, buses, boxes – round in the direction that keeps the inequality true, which can mean rounding up: 125 people in 30-seat buses need x ≥ 4.17, so 5 buses.
Interval notation
| Inequality | Interval | Number line |
|---|---|---|
x < 3 | (-\infty, 3) | open circle at 3, shade left |
x \ge -2 | [-2, \infty) | filled circle at −2, shade right |
-2 < x \le 3 | (-2, 3] | open at −2, filled at 3 |
x \le -2 \text{ or } x \ge 2 | (-\infty, -2] \cup [2, \infty) | two rays |
Quadratic and rational inequalities: the sign chart
- Move everything to one side so the other side is 0 (never multiply by an expression with x – its sign can change).
- Factor, and find the critical points: the zeros of the top, and of the bottom for a fraction.
- They split the number line into intervals. Test one value in each to find the sign.
- Keep the intervals with the sign you need. Include zeros of the top for ≤ or ≥; never include zeros of the bottom.
Absolute value inequalities
|E| < c means E is less than c away from 0: -c < E < c. |E| > c means E is more than c away: E < -c or E > c. If c is negative, |E| < c has no solution and |E| > c is true for every x.
Frequently asked questions
Why can’t I multiply both sides by the denominator?
Because its sign depends on x: where it is negative the inequality would have to flip. Use a sign chart for the whole fraction instead.
What does an open or closed circle mean on the number line?
A closed (filled) circle includes the endpoint (≤ or ≥); an open circle leaves it out (< or >).