Inequalities with Square Roots: What They Are and How to Solve Them

An inequality with a square root has the unknown x inside a radical, under the root sign. In practice, what sits under the radical is not just a number but an expression with x, and that changes how the comparison between the two sides works compared with an ordinary inequality.

The existence condition

When the radical has an even index (square root, fourth root, and so on), before comparing the two sides at all you need to make sure the radical makes sense in the first place. The condition to impose is that the radicand, the expression under the root, is greater than or equal to zero: that is the existence condition, and it comes before any comparison between the two sides. Skip this first step and values of x for which the radical isn’t even defined can slip in as solutions.

The less-than case

When the inequality has the form “radical less than the right-hand side” (or less than or equal to), three conditions are needed together, in a single system: the radicand must be non-negative (the existence condition above), the right-hand side must be positive, and the radicand must be less than the square of the right-hand side. If even one of these three conditions fails, that stretch of x is not part of the solution.

Worked example: take the inequality with the square root of 5x-4 less than x. The three conditions to put into a system are: 5x-4 non-negative, x positive, and 5x-4 less than the square of x. Solving the system, the solution is x greater than or equal to 4/5 and less than 1, or x greater than 4.

The greater-than case

When the inequality has the form “radical greater than the right-hand side” (or greater than or equal to), the reasoning splits into two paths that are then joined together. If the right-hand side is negative, the inequality is automatically satisfied wherever the existence condition on the radicand alone holds, because a radical, when it exists, is never negative. If instead the right-hand side is not negative, both sides must be squared and compared. The final solution is the union of the solutions found in the two cases, not their intersection: any point where the inequality holds in at least one of the two systems is part of the solution.

Worked example: with the inequality square root of 5x+9 greater than x-1, the first system (right-hand side negative) gives -9/5 less than or equal to x less than 1; the second system (right-hand side non-negative, with squaring) gives 1 less than or equal to x less than 8. Joining the two solutions gives -9/5 less than or equal to x less than 8.

The odd index, the simpler case

With an odd index, cube root, fifth root, and so on, there is no existence condition to set: a radical with an odd index is defined for every real number, even when the radicand is negative. Both sides can then be raised directly to the matching power, keeping the inequality’s direction unchanged, with no need to split into separate systems. The same principle holds for equations with a radical, where raising both sides to a power to clear the root is the key step; the difference is that inequalities also require keeping track of the sign of both sides.

Anyone who has already reviewed how equations are solved, radicals included, will find the same underlying idea applied here to a comparison rather than an equality; and anyone who wants to revisit the basics of working with letters first can start from how symbolic notation works in algebra. To place these numbers in the right set (rational, irrational, real), the overview of number sets is a useful stop too.