Theory

Irrational equations and inequalities

The domain conditions of roots and the systems that solve equations and inequalities with roots of even and odd index.

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Domain conditions
The equation √A = B

Domain conditions

Domain conditions for roots of even index

The expression $A$ , like every root of even index, is meaningful only if $A \geq 0$ , and its value is by definition the nonnegative number whose square is $A$ ; so $A \geq 0$ at every point where the root is defined. For example, $x - 2$ is defined for $x \geq 2$ , and $9 = 3$ , not $\pm 3$ . In the solving systems of this chapter the condition $A \geq 0$ appears explicitly or is a consequence of the other conditions, and in either case it must be checked.

The equation √A = B

Equation of the form √A = B

A = B ⟺ {B \geq 0 A = B^{2}

The left-hand side is not negative, so the equation requires $B \geq 0$ ; with both sides nonnegative, squaring preserves the equality in both directions. The domain condition $A \geq 0$ need not be added, because from $A = B^{2}$ it follows that $A \geq 0$ .

Example

$x + 3 = x + 1$ :

{x + 1 \geq 0 x + 3 = (x + 1)^{2}

The second equation is $x^{2} + x - 2 = 0$ , with roots $x = 1$ and $x = - 2$ . Since $- 2 + 1 = - 1 < 0$ , the value $x = - 2$ does not satisfy the first condition and is discarded; the solution is $x = 1$ , and indeed $4 = 2 = 1 + 1$ .

Warning

Squaring both sides without the condition $B \geq 0$ introduces extraneous solutions. From $x = x - 2$ , squaring gives $x = x^{2} - 4 x + 4$ , that is, $x^{2} - 5 x + 4 = 0$ , with roots $x = 1$ and $x = 4$ . For $x = 1$ the left-hand side equals $1 = 1$ and the right-hand side equals $1 - 2 = - 1$ , so $x = 1$ does not solve the original equation; the only solution is $x = 4$ . Squaring cancels signs, therefore the equality between the squares does not guarantee the equality between the two sides.

The theory continues with 4 more sections (the equation with two roots, the inequality a b, the inequality a b 1…), plus all worked examples, the solutions and the PDF.

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