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Questions and Answers
What is the first step to solve the inequality $ rac{x+3}{x^{2}-4} ge 0$?
Which of the following values will make $x^{2}-4$ equal to zero?
What intervals will the expression $ rac{x+3}{x^{2}-4}$ be positive?
Which inequality correctly represents the solution to $ rac{x+3}{x^{2}-4} ge 0$?
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What is the solution set for the inequality $ rac{x+3}{x^{2}-4} ge 0$?
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Study Notes
Solving Rational Inequalities
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The first step in solving the inequality $\frac{x+3}{x^2-4} \ge 0$ is to find the critical points. These are the values of x that make the numerator or denominator equal to zero.
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To find the values of x that make the denominator $x^2-4$ equal to zero, we factor it as $(x+2)(x-2)$. This means $x^2-4$ will equal zero when $x=-2$ or $x=2$.
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The critical points divide the number line into intervals where the expression $\frac{x+3}{x^2-4}$ is either positive or negative. We need to find the intervals where the expression is positive.
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We can use a sign chart to determine the intervals where the expression is positive. We start by placing the critical points on the number line.
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Next, we choose a test point in each interval and evaluate the expression at that point. If the expression is positive, the interval is part of the solution. If the expression is negative, the interval is not part of the solution.
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The inequality that correctly represents the solution to $\frac{x+3}{x^2-4} \ge 0$ is (-3 ≤ x < -2) or (x > 2). Note that the values -2 and 2 are excluded from the solution set because they make the denominator zero, which is undefined.
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Therefore, the solution set for the inequality $\frac{x+3}{x^2-4} \ge 0$ is {(-3, -2) ∪ (2, ∞)}.
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Description
Test your knowledge on solving inequalities with rational expressions. This quiz focuses on understanding the steps to solve the inequality involving the expression (x+3)/(x²-4). You'll also explore intervals of positivity and the solution set for the given inequality.
