Linear Inequalities with Fractions

Table of Contents

1. The Strategy: Eliminate Fractions
2. Example 1: Simple Fraction
3. Example 2: Two Fractions with the Same Denominator
4. Example 3: Different Denominators
5. Example 4: Fraction on Both Sides
6. Example 5: Negative Denominator
7. Example 6: Complex Fraction Inequality
8. Common Mistakes
9. Practice Exercises
10. Related Articles

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1. The Strategy: Eliminate Fractions

The easiest way to solve an inequality with fractions is to multiply both sides by the least common denominator (LCD). This removes all fractions and turns the problem into a simpler inequality.

Important: If the LCD is positive (which it almost always is for constant denominators), the inequality direction does not change. If you ever need to multiply by a negative LCD, reverse the inequality sign.

Steps:

1. Find the LCD of all fractions in the inequality
2. Multiply every term on both sides by the LCD
3. Simplify and solve the resulting inequality

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2. Example 1: Simple Fraction

Solve: $\frac{x}{3}>4$

Step 1: The LCD is 3. Multiply both sides by 3 (positive, direction unchanged): Check: Let $x=15$: $\frac{15}{3}=5>4$ $✓$ Solution: $x\in (12,\infty )$

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3. Example 2: Two Fractions with the Same Denominator

Solve: $\frac{2x+1}{5}\le 3$

Step 1: Multiply both sides by 5: Step 2: Subtract 1: Step 3: Divide by 2: Check: Let $x=7$: $\frac{2\cdot 7+1}{5}=\frac{15}{5}=3\le 3$ $✓$ Solution: $x\in (-\infty ,7]$

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4. Example 3: Different Denominators

Solve: $\frac{x}{2}-\frac{x}{3}>1$

Step 1: The LCD of 2 and 3 is 6. Multiply every term by 6: Step 2: Simplify: Check: Let $x=12$: $\frac{12}{2}-\frac{12}{3}=6-4=2>1$ $✓$ Solution: $x\in (6,\infty )$

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5. Example 4: Fraction on Both Sides

Solve: $\frac{x+1}{4}\ge \frac{x-2}{3}$

Step 1: The LCD of 4 and 3 is 12. Multiply every term by 12: Step 2: Expand: Step 3: Subtract $4x$ from both sides: Step 4: Subtract 3: Step 5: Multiply by $-1$ (reverse the sign!): Check: Let $x=8$:

• Left: $\frac{8+1}{4}=\frac{9}{4}=2.25$
• Right: $\frac{8-2}{3}=\frac{6}{3}=2$
• $2.25\ge 2$ $✓$

Solution: $x\in (-\infty ,11]$

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6. Example 5: Negative Denominator

Solve: $\frac{3x}{-2}<9$

Caution: We can rewrite this as $\frac{3x}{-2}<9$. To eliminate the fraction, multiply both sides by $-2$. Since $-2$ is negative, we must reverse the inequality.

Step 1: Multiply both sides by $-2$ (reverse the sign!): Step 2: Divide by 3: Check: Let $x=0$: $\frac{3\cdot 0}{-2}=0<9$ $✓$ Solution: $x\in (-6,\infty )$

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7. Example 6: Complex Fraction Inequality

Solve: $\frac{2x-1}{3}+\frac{x+4}{6}\le \frac{5}{2}$

Step 1: The LCD of 3, 6, and 2 is 6. Multiply every term by 6: Step 2: Expand: Step 3: Combine like terms: Step 4: Subtract 2: Step 5: Divide by 5: Check: Let $x=2$: $\frac{2\cdot 2-1}{3}+\frac{2+4}{6}=\frac{3}{3}+\frac{6}{6}=1+1=2\le \frac{5}{2}=2.5$ $✓$ Solution: $x\in \left(-\infty ,\frac{13}{5}\right]$

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8. Common Mistakes

Mistake 1: Multiplying Only One Side by the LCD

You must multiply every term on both sides by the LCD.

Wrong: $\frac{x}{2}+3>5\Rightarrow x+3>5$

Correct: $\frac{x}{2}+3>5\Rightarrow x+6>10$

Mistake 2: Forgetting to Distribute After Multiplying

Not $3x+1$!

Mistake 3: Forgetting Sign Reversal with Negative Denominators

When multiplying by a negative number to clear a fraction, always flip the inequality sign.

Mistake 4: Wrong LCD

For $\frac{x}{4}$ and $\frac{x}{6}$, the LCD is 12, not 24. Using a larger common multiple works but creates bigger numbers unnecessarily.

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Practice Exercises

• Inequalities - With Fractions - Practice fraction inequalities
• Inequalities - Both Sides - Reinforce moving terms
• Inequalities - Mixed - Mixed practice

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Related Articles

• Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
• Linear Inequalities - With Parentheses - Expanding brackets
• Linear Inequalities - Simple Inequalities - Review basic types first
• Linear Inequalities - Rules and Formulas - Quick reference card
• Linear Inequalities - Special Cases - No solution or all real numbers