Linear Inequalities - Variable on Both Sides

Table of Contents

1. The Strategy
2. Example 1: Basic Case
3. Example 2: Negative Result
4. Example 3: Negative Coefficient After Collecting
5. Example 4: Larger Coefficients
6. Tips and Reminders
7. Practice Exercises
8. Related Articles

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1. The Strategy

When the variable appears on both sides of the inequality, follow this plan:

1. Move all terms containing $x$ to one side (usually the left)
2. Move all constant terms to the other side
3. Combine like terms
4. Divide by the coefficient of $x$ (flip the sign if it is negative!)

Key idea: The process is the same as solving an equation with the variable on both sides --- just remember the sign-reversal rule when dividing by a negative.

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2. Example 1: Basic Case

Solve: $5x-3>2x+9$

Step 1: Subtract $2x$ from both sides to collect $x$-terms on the left Step 2: Add 3 to both sides to isolate the $x$-term Step 3: Divide by 3 (positive, direction unchanged) Check: Let $x=5$:

• Left side: $5\cdot 5-3=22$
• Right side: $2\cdot 5+9=19$
• $22>19$ $✓$

Solution: $x\in (4,\infty )$

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3. Example 2: Negative Result

Solve: $3x+7\le 6x+1$

Step 1: Subtract $6x$ from both sides Step 2: Subtract 7 from both sides Step 3: Divide by $-3$ (negative, reverse the sign!) Check: Let $x=3$:

• Left side: $3\cdot 3+7=16$
• Right side: $6\cdot 3+1=19$
• $16\le 19$ $✓$

Check boundary $x=2$:

• Left: $3\cdot 2+7=13$
• Right: $6\cdot 2+1=13$
• $13\le 13$ $✓$

Solution: $x\in [2,\infty )$

Alternative approach: You could also move $x$-terms to the right side to avoid a negative coefficient. Subtracting $3x$ from both sides gives $7\le 3x+1$, then $6\le 3x$, then $2\le x$, which is the same as $x\ge 2$.

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4. Example 3: Negative Coefficient After Collecting

Solve: $2x+10>8x-2$

Step 1: Subtract $8x$ from both sides Step 2: Subtract 10 from both sides Step 3: Divide by $-6$ (negative, reverse the sign!) Check: Let $x=0$:

• Left: $2\cdot 0+10=10$
• Right: $8\cdot 0-2=-2$
• $10>-2$ $✓$

Solution: $x\in (-\infty ,2)$

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5. Example 4: Larger Coefficients

Solve: $7x-15\ge 3x+9$

Step 1: Subtract $3x$ from both sides Step 2: Add 15 to both sides Step 3: Divide by 4 (positive, direction unchanged) Check: Let $x=6$:

• Left: $7\cdot 6-15=27$
• Right: $3\cdot 6+9=27$
• $27\ge 27$ $✓$

Let $x=10$:

• Left: $7\cdot 10-15=55$
• Right: $3\cdot 10+9=39$
• $55\ge 39$ $✓$

Solution: $x\in [6,\infty )$

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6. Tips and Reminders

Tip 1: You can choose which side to collect $x$-terms on. If you move them to the side with the larger coefficient, you avoid a negative coefficient and do not need to flip the sign.

Tip 2: Always check your answer by substituting a value from the solution set AND a value outside it. Both checks should confirm the result.

Tip 3: Watch the signs carefully when moving terms. When a term crosses the inequality sign, its sign changes (positive becomes negative and vice versa).

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Practice Exercises

• Inequalities - Both Sides - Practice these types of inequalities
• Inequalities - addition and subtraction - Review simpler problems first
• Inequalities - Mixed - Mixed practice

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Related Articles

• Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
• Linear Inequalities - Simple Inequalities - Simpler types to master first
• Linear Inequalities - With Parentheses - Next level: expanding brackets
• Linear Inequalities - Rules and Formulas - Quick reference card