☰ Contents · Algebra

Inequalities and equations with powers

Lessons 13 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
13

Inequalities and equations with powers

Textbook: pp. 51–55
GoalUse the increase of power functions to solve simple inequalities with powers; solve irrational equations and inequalities and check for extraneous roots.
New words
irrational equation · irratsional tenglamaextraneous root · chet ildizsquaring both sides · ikkala qismini kvadratga ko‘tarisharithmetic root · arifmetik ildiz
Explanation

We use the increase of power functions in inequalities. For odd n the function y = xⁿ increases on the whole line, so xⁿ > a gives x > ⁿ√a. For even n the values of y = xⁿ depend on |x|: x⁴ < 16 gives |x| < 2, i.e. −2 < x < 2. An equation with the unknown under a root sign is called an irrational equation. To solve it we square both sides, solve the new equation and always check: squaring may produce extraneous roots that do not satisfy the original equation. Since an arithmetic root is non-negative, the equation √f = (negative number) has no roots. For an inequality with a root, first find the values of x for which the root makes sense (the domain), then square both sides.

Worked examples
x³ > 64 gives x > 4 (the cube function increases). x⁴ ≤ 256 gives |x| ≤ 4, i.e. −4 ≤ x ≤ 4.
√(x + 6) = x: squaring gives x + 6 = x², x² − x − 6 = 0, x₁ = 3, x₂ = −2. Check: for x = 3, √9 = 3, true; for x = −2, √4 = 2 ≠ −2, an extraneous root. Answer: x = 3.
Class activity

“Extraneous root hunters”: teams square an irrational equation and find the roots, then put each root back into the original and declare it “stays” or “extraneous”. One point for each correct check.

Practice
1
Solve x⁵ ≥ 243.
2
Solve x⁴ > 256.
3
Solve √(3x − 2) = 4.
4
Why does √x + 4 = 0 have no roots?