Rational Inequalities

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Idea

A rational expression can change sign at a numerator zero or at a denominator zero. The latter is a forbidden input, even if factors cancel later. Start by recording the original domain.

Method

Move all terms to one side, form one fraction, factor numerator and denominator, and test intervals separated by every critical point. Include eligible numerator zeros for non-strict inequalities; never include denominator zeros.

Worked example

Solve $(x-1)/(x+2)\ge0$ . Exclude $-2$ , and split at $-2,1$ . Testing $-3,0,2$ gives positive, negative, positive. Thus

x\in(-\infty,-2)\cup[1,\infty).

At $1$ the expression is zero; at $-2$ it is undefined.

Common mistake

Cross-multiplying by $x+2$ without knowing its sign can reverse the inequality in part of the domain and produce the wrong answer.

Check your understanding

Solve $(x+1)/(x-2)<0$ .

Show answer

$(-1,2)$ . The numerator zero is excluded because the inequality is strict; the denominator zero is excluded by the domain.

Explore

Sign of a rational expression

Try this. Move the test point across −2 and 1. A zero may be included in a non-strict inequality; a denominator zero never can.

Test x0.5

(x − 1)/(x + 2) at x = 0.5: -0.2 · negative. Positive on (−∞, −2) ∪ (1, ∞); x = −2 is always excluded.

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

1/(x-2)>0

x > 2

x ≥ 2

x < 2

Hint 1 · Find a starting point

The numerator is positive. What sign must the denominator have?

Hint 2 · Take the next step

Do not include the point where the denominator is zero.

Show the reasoning

Answer: x > 2

The fraction is positive exactly when x−2>0; x=2 is undefined.

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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