Inequalities and Their Solutions

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Questions and Answers

What values of $x$ make the inequality $\frac{x}{x^2-1} < 0$ true?

$x < -1$ or $-1 < x < 1$ or $x > 1$

Identify the intervals for which the inequality $\frac{x^2-5x+6}{x^2-5x-6} > 0$ holds true.

$(-\infty, 2) \cup (3, 4) \cup (5, \infty)$

For the inequality $\frac{6x^2+x-1}{20x^2-x-1} \le 0$, what critical points must be considered?

$x = -\frac{1}{3}, \frac{1}{2}, \frac{6}{5}$

What is the significance of determining inequalities like $\frac{x^2-5x+6}{x^2-5x-6} > 0$ in practical applications?

<p>It helps identify ranges for $x$ where a function behaves favorably, such as maximizing profit or minimizing cost.</p> Signup and view all the answers

Describe how to find the solution set for $\frac{6x^2+x-1}{20x^2-x-1} \le 0$.

<p>Factor the numerator and denominator, identify critical points, and test intervals to find where the inequality holds.</p> Signup and view all the answers

Flashcards

Rational Inequality

A rational inequality is an inequality where the variable appears in the denominator of a fraction. To solve them, we need to find the critical points (where the numerator or denominator is zero or undefined) and then analyze the sign of the expression in the intervals defined by these critical points.

Finding Critical Points

Factoring the numerator and denominator of the inequality, the critical points are the values of x that make each factor equal to zero. These points divide the number line into intervals.

Testing Intervals

Once we have the critical points, we test the sign of the expression in each interval. We choose a test value within each interval and evaluate the expression. The sign of the expression tells us whether the inequality is satisfied in that interval.

Solution of Rational Inequality

The solution to the rational inequality is the set of intervals where the expression satisfies the inequality. We need to consider the original inequality's sign and whether the critical points should be included or excluded.

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Solving Rational Inequalities - Example

In problems like 'x / (x^2-1) < 0', we need to solve for x, considering the critical points and the sign of the expression in each interval. The solution is the set of x values that make the inequality true.

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