Solving Inequalities

Inequalities are like equations, but instead of saying two things are equal, they show that one is greater than or less than the other. You'll learn to solve and graph inequalities just like you solve equations!

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What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using inequality symbols.

Inequality Symbols:

• < means "less than" (3 < 5)
• > means "greater than" (7 > 2)
• ≤ means "less than or equal to" (x ≤ 10)
• ≥ means "greater than or equal to" (y ≥ -3)

Examples:

• x > 5 (x is greater than 5)
• y ≤ 10 (y is less than or equal to 10)
• 2x + 3 < 15 (an inequality to solve)

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Inequalities vs. Equations

Equation: x + 3 = 7 (has ONE solution: x = 4)

Inequality: x + 3 > 7 (has MANY solutions: x > 4)

• Solutions: 5, 6, 7, 100, 4.1, etc. (infinitely many!)

Key Difference: Inequalities have a range of solutions!

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Solving One-Step Inequalities

Solve inequalities just like equations - use inverse operations!

Addition and Subtraction

Example 1: Solve x + 5 > 12

Step 1: Subtract 5 from both sides x + 5 - 5 > 12 - 5 x > 7

Answer: x > 7

Meaning: Any number greater than 7 is a solution!

• Test: x = 8 → 8 + 5 = 13 > 12 ✓
• Test: x = 10 → 10 + 5 = 15 > 12 ✓

Example 2: Solve y - 3 ≤ 8

Step 1: Add 3 to both sides y - 3 + 3 ≤ 8 + 3 y ≤ 11

Answer: y ≤ 11

Meaning: Any number less than or equal to 11 works!

Multiplication and Division (Positive Numbers)

Example 3: Solve 4x < 20

Step 1: Divide both sides by 4 4x ÷ 4 < 20 ÷ 4 x < 5

Answer: x < 5

Example 4: Solve x/3 ≥ 6

Step 1: Multiply both sides by 3 3 × (x/3) ≥ 3 × 6 x ≥ 18

Answer: x ≥ 18

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THE SPECIAL RULE: Multiplying or Dividing by Negatives

CRITICAL: When you multiply or divide both sides by a NEGATIVE number, you must FLIP the inequality symbol!

Why? Think about it: 5 > 2 is true

• Multiply both by -1: -5 ? -2
• On a number line, -5 is LEFT of -2, so -5 < -2
• The inequality flipped!

Example 1: Dividing by a Negative

Solve: -3x > 12

Step 1: Divide both sides by -3 AND flip the symbol -3x ÷ (-3) < 12 ÷ (-3) (> becomes <) x < -4

Answer: x < -4

Check: Try x = -5

• -3(-5) = 15 > 12 ✓

Example 2: Multiplying by a Negative

Solve: -x/2 ≤ 4

Step 1: Multiply both sides by -2 AND flip the symbol (-2) × (-x/2) ≥ (-2) × 4 (≤ becomes ≥) x ≥ -8

Answer: x ≥ -8

Example 3: Watch the Sign!

Solve: -5y < 25

Step 1: Divide by -5, flip the symbol -5y ÷ (-5) > 25 ÷ (-5) y > -5

Answer: y > -5

Remember: ONLY flip when multiplying or dividing by a NEGATIVE!

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Solving Multi-Step Inequalities

Use the same strategy as multi-step equations!

Example 1: Two Steps

Solve: 3x + 7 < 22

Step 1: Subtract 7 from both sides 3x < 15

Step 2: Divide by 3 x < 5

Answer: x < 5

Example 2: With Negative Coefficient

Solve: -2x + 5 ≥ 13

Step 1: Subtract 5 from both sides -2x ≥ 8

Step 2: Divide by -2 AND FLIP x ≤ -4

Answer: x ≤ -4

Example 3: Distributive Property

Solve: 3(x - 4) > 15

Step 1: Distribute 3x - 12 > 15

Step 2: Add 12 to both sides 3x > 27

Step 3: Divide by 3 x > 9

Answer: x > 9

Example 4: Variable on Both Sides

Solve: 5x - 3 < 2x + 9

Step 1: Subtract 2x from both sides 3x - 3 < 9

Step 2: Add 3 to both sides 3x < 12

Step 3: Divide by 3 x < 4

Answer: x < 4

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Graphing Inequalities on a Number Line

Symbols and Graphing:

• < or > use an open circle (○) - the number is NOT included
• ≤ or ≥ use a closed/filled circle (●) - the number IS included
• Shade to the left for "less than"
• Shade to the right for "greater than"

Example 1: x > 3

Graph: Open circle at 3, shade RIGHT

• 3 is NOT included (open circle)
• All numbers greater than 3 are shaded

Example 2: x ≤ -2

Graph: Closed circle at -2, shade LEFT

• -2 IS included (closed circle)
• All numbers less than or equal to -2 are shaded

Example 3: x ≥ 0

Graph: Closed circle at 0, shade RIGHT

• 0 IS included
• All positive numbers and zero are shaded

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Writing Inequalities from Graphs

Look at the circle (open or closed) and the shading direction!

Example 1: Closed circle at 5, shaded right

• Answer: x ≥ 5

Example 2: Open circle at -3, shaded left

• Answer: x < -3

Example 3: Closed circle at 0, shaded left

• Answer: x ≤ 0

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Writing Inequalities from Word Problems

Translate words into inequality symbols!

Word Clues:

Greater than (>):

• "more than"
• "exceeds"
• "above"

Greater than or equal to (≥):

• "at least"
• "no less than"
• "minimum"

Less than (<):

• "fewer than"
• "below"
• "under"

Less than or equal to (≤):

• "at most"
• "no more than"
• "maximum"

Example 1: At Least

"You must be at least 13 years old to have a social media account."

Inequality: a ≥ 13 (age must be greater than or equal to 13)

Example 2: No More Than

"The elevator can hold no more than 2,000 pounds."

Inequality: w ≤ 2,000 (weight must be less than or equal to 2,000)

Example 3: More Than

"Sarah has more than $50 in her account."

Inequality: m > 50 (money is greater than 50)

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Real-World Applications

Shopping with a Budget

Problem: You have $100 to spend. You buy a shirt for$25. If jeans cost $35 each, how many pairs can you buy?

Let x = number of jeans

Inequality: 25 + 35x ≤ 100

Solution: 35x ≤ 75 x ≤ 75/35 x ≤ 2.14...

Answer: You can buy at most 2 pairs of jeans (since you can't buy a fraction of jeans!)

Grade Requirements

Problem: Your test average must be at least 80% to get a B. You've taken 3 tests and scored 75, 82, and 78. What do you need on the 4th test?

Let x = 4th test score

Inequality: (75 + 82 + 78 + x)/4 ≥ 80

Solution: (235 + x)/4 ≥ 80 235 + x ≥ 320 x ≥ 85

Answer: You need at least 85% on the 4th test

Speed Limits

Problem: The speed limit is 65 mph. Write an inequality for legal speeds.

Let s = speed

Inequality: s ≤ 65

Weight Restrictions

Problem: A bridge has a maximum weight of 10 tons. A truck weighs 3 tons. If each crate weighs 0.5 tons, how many crates can the truck carry?

Let c = number of crates

Inequality: 3 + 0.5c ≤ 10

Solution: 0.5c ≤ 7 c ≤ 14

Answer: Maximum 14 crates

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Compound Inequalities (Introduction)

Sometimes a value is between two numbers!

Example: "Temperature should be between 60°F and 75°F"

Written as: 60 ≤ t ≤ 75 or 60 < t < 75

This means t ≥ 60 AND t ≤ 75 at the same time.

Graph: Closed circles at 60 and 75, shade between them

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Common Mistakes to Avoid

❌ Mistake 1: Forgetting to flip when dividing by negative

• Wrong: -2x > 8 → x > -4
• Right: -2x > 8 → x < -4 (flip!)

❌ Mistake 2: Flipping when you shouldn't

• Wrong: 3x < 12 → x > 4 (no negative, don't flip!)
• Right: 3x < 12 → x < 4

❌ Mistake 3: Using wrong circle on graph

• < or > : open circle ○
• ≤ or ≥ : closed circle ●

❌ Mistake 4: Shading wrong direction

• < or ≤ : shade LEFT

• or ≥ : shade RIGHT

❌ Mistake 5: Treating inequality like equation

• Inequalities have MANY solutions, not just one!

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Practice Strategy

Step 1: Solve like an equation

• Use inverse operations
• Isolate the variable

Step 2: Watch for negatives!

• If you multiply or divide by a negative, FLIP the symbol

Step 3: Graph your answer

• Choose correct circle (open or closed)
• Shade correct direction

Step 4: Check with a test value

• Pick a number that should work
• Substitute it in
• Verify it makes the inequality true

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Checking Your Solution

Example: Solve x + 5 < 12, got answer x < 7

Check 1: Try x = 6 (should work)

• 6 + 5 = 11 < 12 ✓ True!

Check 2: Try x = 7 (boundary, should NOT work since it's <, not ≤)

• 7 + 5 = 12 NOT < 12 ✓ Correct!

Check 3: Try x = 8 (should NOT work)

• 8 + 5 = 13 NOT < 12 ✓ Correct!

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Quick Reference Chart

| Operation | Example | Rule | |-----------|---------|------| | Add/Subtract | x + 3 > 7 → x > 4 | Same as equations | | Multiply/Divide (positive) | 2x < 8 → x < 4 | Same as equations | | Multiply/Divide (negative) | -2x < 8 → x > -4 | FLIP the symbol! |

Graphing:

• < or > : Open circle, shade direction
• ≤ or ≥ : Closed circle, shade direction

Word Clues:

• At least, minimum, no less than → ≥
• At most, maximum, no more than → ≤
• More than, above, exceeds → >
• Less than, below, under → <

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Summary

Inequalities show relationships using <, >, ≤, ≥ symbols.

Solving:

• Use inverse operations like equations
• FLIP the symbol when multiplying or dividing by a negative
• Solution is a range of values, not just one number

Graphing:

• Open circle (○) for < or >
• Closed circle (●) for ≤ or ≥
• Shade left for "less than"
• Shade right for "greater than"

Applications:

• Budgets (at most, no more than)
• Requirements (at least, minimum)
• Limits (maximum, speed limits)
• Constraints (weight, capacity)

Master inequalities and you're ready for systems of inequalities, linear programming, and real-world optimization!