Quadratic Inequalities

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Linear Inequalities: Solve Like an Equation

A linear inequality solves exactly like a linear equation: $x$ terms on one side, numbers on the other. Two differences only —

The answer is a range of values, not a single value.
If you multiply or divide both sides by a negative number, the inequality sign flips.

Reflex: Multiply or divide by a negative? Flip the inequality sign.

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4 worked examples

A linear inequality solves exactly like a linear equation: $x$ terms on one side, numbers on the other. Two differences only —

The answer is a range of values, not a single value.
If you multiply or divide both sides by a negative number, the inequality sign flips.

Reflex: Multiply or divide by a negative? Flip the inequality sign.

\begin{aligned}> 0 &: \ \text{take the part above the } x\text{-axis}\\< 0 &: \ \text{take the part below}\end{aligned}

Simplify, zero one side, factorize to find the roots — then sketch the parabola through the roots and read off the region that matches the sign.

Reflex: The roots are only the boundaries — the answer is the region you read off the sketch.

-6x^2 + 5x - 1 > 0 \;\;\xrightarrow{\;\div\,(-1)\;}\;\; 6x^2 - 5x + 1 < 0

A negative $x^2$ term (the curve opens downwards) is awkward to factorize — divide by $-1$ to make it positive.

Reflex: Dividing by −1 flips the inequality sign.

x < -4 \ \text{ or } \ x > 2 \;\;\Longrightarrow\;\; (x+4)(x-2) > 0

Work backwards: solution → factors → expand → match coefficients. Reflex: Outside roots rebuilds with $>0$ ; between roots rebuilds with $<0$ .