1Cademy - Try It: Solving $$\frac{1}{2} + \frac{4}{x^2}

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Example: Solving a Rational Inequality Requiring a Common Denominator

Example

Try It: Solving $\frac{1}{2} + \frac{4}{x^2} < \frac{3}{x}$

To solve the rational inequality $\frac{1}{2} + \frac{4}{x^2} < \frac{3}{x}$ , first move all terms to the left side by subtracting $\frac{3}{x}$ to obtain $\frac{1}{2} + \frac{4}{x^2} - \frac{3}{x} < 0$ . Find the least common denominator (LCD), which is $2x^2$ , and rewrite each fraction: $\frac{x^2}{2x^2} + \frac{8}{2x^2} - \frac{6x}{2x^2} < 0$ . Combine the fractions and rearrange the numerator in descending order: $\frac{x^2 - 6x + 8}{2x^2} < 0$ . Factor the numerator to get $\frac{(x - 2)(x - 4)}{2x^2} < 0$ . Find the zero partition numbers by setting the numerator and denominator to zero: the numerator yields $x = 2$ and $x = 4$ , and the denominator yields $x = 0$ . These partition numbers divide the number line into four intervals: $(-\infty, 0)$ , $(0, 2)$ , $(2, 4)$ , and $(4, \infty)$ . Testing values in each interval shows that the quotient is negative only in the interval $(2, 4)$ . Since the inequality is strictly less than $0$ , the partition numbers are excluded. Written in interval notation, the solution is $(2, 4)$ .

Updated 2026-05-26

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OpenStax Intermediate Algebra

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