Linear Inequalities with Parentheses

Table of Contents

1. The Distributive Law
2. Example 1: Simple Parentheses
3. Example 2: Parentheses on Both Sides
4. Example 3: Negative Sign Before Brackets
5. Example 4: Negative Multiplier Before Brackets
6. Example 5: Multiple Parentheses
7. Common Mistakes
8. Practice Exercises
9. Related Articles

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1. The Distributive Law

Before you can solve an inequality with parentheses, you need to expand them using the distributive law:

Important: When the number in front of the brackets is negative, pay extra attention to the signs of each term inside the brackets.

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2. Example 1: Simple Parentheses

Solve: $3(x+2)>15$

Step 1: Expand the brackets Step 2: Subtract 6 from both sides Step 3: Divide by 3 Check: Let $x=4$: $3(4+2)=3\cdot 6=18>15$ $✓$ Solution: $x\in (3,\infty )$

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3. Example 2: Parentheses on Both Sides

Solve: $2(x+4)\le 3(x-1)$

Step 1: Expand both sides Step 2: Subtract $3x$ from both sides Step 3: Subtract 8 from both sides Step 4: Multiply by $-1$ (reverse the sign!) Check: Let $x=12$:

• Left: $2(12+4)=2\cdot 16=32$
• Right: $3(12-1)=3\cdot 11=33$
• $32\le 33$ $✓$

Check boundary $x=11$:

• Left: $2(11+4)=30$
• Right: $3(11-1)=30$
• $30\le 30$ $✓$

Solution: $x\in [11,\infty )$

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4. Example 3: Negative Sign Before Brackets

Solve: $10-(x+3)>2$

Step 1: Expand the brackets (the minus sign distributes to both terms) Step 2: Simplify the left side Step 3: Subtract 7 from both sides Step 4: Multiply by $-1$ (reverse the sign!) Check: Let $x=3$: $10-(3+3)=10-6=4>2$ $✓$ Solution: $x\in (-\infty ,5)$

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5. Example 4: Negative Multiplier Before Brackets

Solve: $-4(2x-1)\ge 20$

Step 1: Expand the brackets

Notice: $-4\cdot 2x=-8x$ and $-4\cdot (-1)=+4$

Step 2: Subtract 4 from both sides Step 3: Divide by $-8$ (reverse the sign!) Check: Let $x=-3$: $-4(2\cdot (-3)-1)=-4(-6-1)=-4\cdot (-7)=28\ge 20$ $✓$ Check boundary $x=-2$: $-4(2\cdot (-2)-1)=-4(-4-1)=-4\cdot (-5)=20\ge 20$ $✓$ Solution: $x\in (-\infty ,-2]$

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6. Example 5: Multiple Parentheses

Solve: $3(x-2)-2(x+5)<1$

Step 1: Expand all brackets Step 2: Combine like terms on the left Step 3: Add 16 to both sides Check: Let $x=10$: $3(10-2)-2(10+5)=24-30=-6<1$ $✓$ Solution: $x\in (-\infty ,17)$

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7. Common Mistakes

Mistake 1: Forgetting to Distribute to All Terms

Correct: $2(x+5)=2x+10$

Mistake 2: Sign Error with Negative Multiplier

Correct: $-3(x-4)=-3x+12$

Remember: negative times negative equals positive!

Mistake 3: Forgetting the Invisible 1

Not $-x+3$!

Mistake 4: Expanding Before Checking

Sometimes it is faster to divide first. For example:

$3(x+2)>15$ can be solved by dividing both sides by 3 directly: $x+2>5$, then $x>3$

Tip: You can divide both sides by a positive number before expanding. This sometimes makes the arithmetic easier.

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Practice Exercises

• Inequalities - With Parentheses - Practice expanding and solving
• 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 - Variable on Both Sides - Prerequisite: moving terms
• Linear Inequalities - With Fractions - Next level: fractions
• Linear Inequalities - Simple Inequalities - Review basic types
• Linear Inequalities - Rules and Formulas - Quick reference card