How to Solve Rational Inequalities?

• Step 1: Write the inequality as a factor on the left side and zero on the right side.
• Step 2: Determine critical points – points where the rational expression is zero or undefined.
• Step 3: Use the critical points to divide the number line into intervals.
• Step 4: Test one value in each interval. Above the number line, show the sign of each factor of the numerator and denominator in each interval. Show the sign of the quotient under the number line.
• Step 5: Determine the intervals for which the inequality is true. Write the solution in interval notation.

Solving Rational Inequalities – Example 1:

Exercises for Solving Rational Inequalities

Solve.

1. \(\color{blue}{\frac{x+4}{2x-5}\le 7}\)
2. \(\color{blue}{\frac{6}{x-8}\le 3}\)
3. \(\color{blue}{\frac{1}{2}x-2\le \:3x}\)

1. \(\color{blue}{\left(-\infty \:\:,\:\frac{5}{2}\right)\cup [3,\:\infty)}\)
2. \(\color{blue}{\left(-\infty \:\:,\:8\right)\cup \:[10,\:\infty)}\)
3. \(\color{blue}{\:[-\frac{4}{5},\:\infty )}\)

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