Quadratic and Higher-Degree Inequalities

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Idea

A polynomial changes sign only at a real zero. Factor it, arrange its zeros on a number line, and test one point in each remaining interval. This gives a complete sign table, not just a collection of sample values.

Method

At a zero of odd multiplicity the sign changes; at even multiplicity it does not. Include zeros for $\le$ or $\ge$ , and exclude them for strict inequalities. The zero-product rule alone does not solve an inequality.

Worked example

Solve $(x-1)(x+2)\le0$ . The zeros divide the line at $-2$ and $1$ . At $x=-3$ , both factors are negative and the product is positive. At $x=0$ , the product is negative. At $x=2$ , both factors are positive. Include the endpoints, giving $[-2,1]$ .

Common mistake

Multiplying or dividing an inequality by a negative number reverses its direction. Dividing by an expression of unknown sign requires cases.

Check your understanding

Solve $(x-3)^2>0$ .

Show answer

All real numbers except $3$ : $(-\infty,3)\cup(3,\infty)$ . A square is never negative.

Explore

Factors and zeros

Try this. Move the roots together and apart. Match the expanded coefficients to their sum and product, then compare touching with crossing the axis.

First root-2Second root2

(x − (-2))(x − (2)) = x² − (0)x + (-4). Zeros: -2, 2. Simple roots: the graph crosses at each zero.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Solve

(x-1)(x+2)>0

x ≥ −2

x < −2 or x > 1

−2 < x < 1

Hint 1 · Find a starting point

The zeros split the number line into three intervals.

Hint 2 · Take the next step

A product is positive when its factors have the same sign.

Show the reasoning

Answer: x < −2 or x > 1

Both factors are negative left of −2 and positive right of 1. Strict inequality excludes the zeros.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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