Sizu Riatto
Algebra · Chapter 15

Solve and Shade an Inequality

“At most” and “at least” are inequalities. Solve them like equations, with one extra rule: dividing by a negative number turns the sign around.

Home › Calculators › Solve and Shade an Inequality

Opens on Chapter 15’s example: Teo’s $60: a $35 game and as many $4 packs as he can afford, 4x + 35 ≤ 60.

a·x + b (sign) c
Enter your numbers, then press Work it out.

How it works

Do the same to both sides, as with an equation: take away the number, then divide by what multiplies the letter. Adding, taking away, or multiplying and dividing by a positive number keeps the sign. Dividing (or multiplying) by a negative turns it around, because it flips the order of the number line: 2 < 3, but −2 > −3. On the line, a filled dot means the boundary is included (≤, ≥) and an open dot means it isn’t (<, >). For things you count, only whole numbers work, so round the boundary in the direction the answer allows, then check the last one in the original.

From the book, Chapter 15. 4x + 35 ≤ 60 gives x ≤ 6.25, so Teo can buy 6 packs and spend $59. Free delivery needs at least 95 sheets; the elevator takes up to 18.2, so 18 boxes; an average of 80 after 72 and 81 needs at least 87. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.