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Compound Inequality Solver

Enter the bounds and coefficients to solve the compound inequality for x.

How it is calculated

The tool solves a < bx + c < d for x. Step one subtracts c from all three parts: a − c < bx < d − c. Step two divides all three by b, giving the two ends (a − c)/b and (d − c)/b. When b is negative, dividing reverses the inequality signs, so the two ends swap and the smaller one is always shown on the left. The inequalities are strict, so the end values themselves are not part of the solution. Answers are shown to four decimals. There is no result when b = 0 (x disappears) or when a is not smaller than d (no x can satisfy the chain).

Worked example

Solve 2 < −3x + 5 < 14, so a = 2, b = −3, c = 5, d = 14. Subtract 5 from every part: −3 < −3x < 9. Dividing by −3 gives (2 − 5)/(−3) = 1 and (14 − 5)/(−3) = −3. Because b is negative the order flips, and the tool shows -3.0000 < x < 1.0000. Check with x = 0: −3·0 + 5 = 5, which lies between 2 and 14, so it works.

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Common mistakes

  • Forgetting to reverse the direction when b is negative. With b = −3 the larger bound ends up on the left unless the signs are flipped.
  • Subtracting c from only the middle part. It has to be removed from the lower bound a and the upper bound d as well.
  • Entering the bounds in the wrong order, such as a larger than d. The chain a < … < d then has no solution and the tool returns nothing.
  • Treating the ends as included. The inequalities are strict, so x equal to either end value does not satisfy the original statement.
  • Setting b = 0 and expecting an answer. With no x term the expression is a constant, so there is nothing to solve for.

About this calculator

A compound inequality of the form a less than bx + c, and bx + c less than d squeezes the expression bx + c between two bounds at once, and it's solved the same way a single inequality is — by isolating x through valid algebraic operations applied to all three parts simultaneously: first subtract c from every part, then divide every part by b.

The one rule that trips people up is that dividing or multiplying an inequality by a negative number flips its direction — so if b is negative, "a less than (something) less than d" becomes "a greater than (something) greater than d" after dividing, and the bounds swap places accordingly. Getting this sign-flip right is exactly what separates a correctly solved compound inequality from a silently wrong one.

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Comments (1)

  • Editorial team

    After solving, test both ends yourself. For the chain 2 below -3x + 5 below 14, the tool gives x between -3.0000 and 1.0000. At x = 1 the middle expression equals exactly 2, and at x = -3 it equals exactly 14, so neither end belongs to the solution, because the inequalities are strict. Then pick any value between them to see the chain hold. Have you met a compound inequality with a non-strict sign that you wanted this tool to handle?

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