How to Solve Rational Inequalities?
Tutor-style math help
Solve Rational Inequalities: what to notice and how to work it
Inequalities skill
Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.
What to notice first
Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.
Common student mistake
Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.
Key formulas and cues
\(a<b\)
\(a\le b\)
\(\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}\)
\(|x-a|<b \Rightarrow a-b<x<a+b\)
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
Example: \(-3x>12\)
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Answer: \(x<-4\)
Keep the sign
Example: \(x+5\le9\)
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Answer: \(x\le4\)
Try one before moving on
Try: Solve \(-2x\le10\).
Answer: \(x\ge-5\). Divide by -2 and flip the sign.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Solve Rational Inequalities: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Solve \(x-2>5\).
2. When must the inequality sign flip?
3. \(x\le3\) uses:
- 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.
- \(\color{blue}{\frac{x+4}{2x-5}\le 7}\)
- \(\color{blue}{\frac{6}{x-8}\le 3}\)
- \(\color{blue}{\frac{1}{2}x-2\le \:3x}\)
- \(\color{blue}{\left(-\infty \:\:,\:\frac{5}{2}\right)\cup [3,\:\infty)}\)
- \(\color{blue}{\left(-\infty \:\:,\:8\right)\cup \:[10,\:\infty)}\)
- \(\color{blue}{\:[-\frac{4}{5},\:\infty )}\)
Original price was: $109.99.$54.99Current price is: $54.99.
Original price was: $109.99.$54.99Current price is: $54.99.
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