1. Solutions to Inequalities

Definition: A solution to an inequality is any value that makes the inequality true.

Inequality Symbols:

Testing Solutions:

To check if a value is a solution, substitute it into the inequality and verify if the statement is true.

Example: Is \( x = 4 \) a solution to \( 2x + 3 < 15 \)?

Substitute: \( 2(4) + 3 < 15 \)

\( 8 + 3 < 15 \)

\( 11 < 15 \) ✓ True, so \( x = 4 \) is a solution

Important: Unlike equations (which usually have one solution), inequalities typically have infinitely many solutions!

2. Graph Inequalities on Number Lines

Circle Types:

• Open Circle (○): Used for \( < \) or \( > \) (value NOT included)
• Closed Circle (●): Used for \( \leq \) or \( \geq \) (value IS included)

Arrow Direction:

• Arrow pointing LEFT (←): For \( < \) or \( \leq \) (less than)
• Arrow pointing RIGHT (→): For \( > \) or \( \geq \) (greater than)

Steps to Graph:

1. Draw a number line
2. Locate the critical value
3. Draw an open or closed circle at that value
4. Draw an arrow in the correct direction

Examples:

Example 1: Graph \( x > 3 \)

• Open circle at 3

• Arrow pointing right →

Example 2: Graph \( x \leq -2 \)

• Closed circle at -2

• Arrow pointing left ←

Example 3: Graph \( x \geq 0 \)

• Closed circle at 0

• Arrow pointing right →

Memory Tip:

"Open circle = Open inequality (no equal sign)"

"Closed circle = Closed inequality (has equal sign)"

3. Write Inequalities from Number Lines

Steps to Write:

1. Identify the critical value (where the circle is)
2. Determine if the circle is open or closed
3. Determine the direction of the arrow
4. Write the inequality using the correct symbol

Quick Reference:

Examples:

Example 1: Closed circle at 5, arrow pointing right

Answer: \( x \geq 5 \)

Example 2: Open circle at -3, arrow pointing left

Answer: \( x < -3 \)

4. Rules for Solving Inequalities

Inequalities follow the same rules as equations, with ONE CRITICAL EXCEPTION:

Rule 1: Addition/Subtraction

Sign stays the same when adding or subtracting the same number from both sides.

If \( a < b \), then \( a + c < b + c \)

If \( a < b \), then \( a - c < b - c \)

Rule 2: Multiplication/Division by a POSITIVE Number

Sign stays the same when multiplying or dividing both sides by a positive number.

If \( a < b \) and \( c > 0 \), then \( ac < bc \)

If \( a < b \) and \( c > 0 \), then \( \frac{a}{c} < \frac{b}{c} \)

Rule 3: Multiplication/Division by a NEGATIVE Number ⚠️

FLIP THE INEQUALITY SIGN when multiplying or dividing both sides by a negative number!

If \( a < b \) and \( c < 0 \), then \( ac > bc \) (sign flips!)

If \( a < b \) and \( c < 0 \), then \( \frac{a}{c} > \frac{b}{c} \) (sign flips!)

Why Does the Sign Flip?

Example: We know \( 3 < 5 \)

Multiply both sides by -1:

\( -3 \) ? \( -5 \)

Since -3 is greater than -5, the inequality becomes: \( -3 > -5 \)

The sign flipped from \( < \) to \( > \)!

Summary Table:

5. Solve One-Step Inequalities

Strategy: Use inverse operations to isolate the variable, just like equations.

Type 1: Addition Inequalities

Form: \( x + a < b \) or \( x + a > b \)

Solution: Subtract \( a \) from both sides

Example: Solve \( x + 5 < 12 \)

Subtract 5: \( x + 5 - 5 < 12 - 5 \)

Answer: \( x < 7 \)

Type 2: Subtraction Inequalities

Form: \( x - a < b \) or \( x - a > b \)

Solution: Add \( a \) to both sides

Example: Solve \( x - 8 \geq 3 \)

Add 8: \( x - 8 + 8 \geq 3 + 8 \)

Answer: \( x \geq 11 \)

Type 3: Multiplication Inequalities (Positive)

Form: \( ax < b \) (where \( a > 0 \))

Solution: Divide by \( a \) (sign stays same)

Example: Solve \( 4x > 20 \)

Divide by 4: \( \frac{4x}{4} > \frac{20}{4} \)

Answer: \( x > 5 \)

Type 4: Multiplication Inequalities (Negative) ⚠️

Form: \( -ax < b \)

Solution: Divide by negative \( a \) and FLIP the sign!

Example: Solve \( -3x \leq 12 \)

Divide by -3 and flip: \( \frac{-3x}{-3} \geq \frac{12}{-3} \)

Answer: \( x \geq -4 \)

Type 5: Division Inequalities

Form: \( \frac{x}{a} < b \)

Solution: Multiply by \( a \) (watch sign if negative!)

Example: Solve \( \frac{x}{5} > 6 \)

Multiply by 5: \( 5 \cdot \frac{x}{5} > 5 \cdot 6 \)

Answer: \( x > 30 \)

6. Solve Two-Step Inequalities

General Form: \( ax + b < c \) or similar variations

Steps to Solve:

1. Step 1: Add or subtract to isolate the variable term
2. Step 2: Multiply or divide to solve for the variable
3. Step 3: Flip the sign if multiplying/dividing by negative

Examples:

Example 1: Solve \( 3x + 5 < 17 \)

Step 1: Subtract 5: \( 3x < 12 \)

Step 2: Divide by 3: \( x < 4 \)

Example 2: Solve \( \frac{x}{2} - 4 \geq 6 \)

Step 1: Add 4: \( \frac{x}{2} \geq 10 \)

Step 2: Multiply by 2: \( x \geq 20 \)

Example 3: Solve \( -2x + 7 > 15 \) (with negative coefficient!)

Step 1: Subtract 7: \( -2x > 8 \)

Step 2: Divide by -2 and flip: \( x < -4 \)

Example 4: Solve \( 5 - 3x \leq 20 \)

Subtract 5: \( -3x \leq 15 \)

Divide by -3 and flip: \( x \geq -5 \)

7. Solve Multi-Step Inequalities

General Strategy:

1. Simplify each side (distribute, combine like terms)
2. Move variable terms to one side
3. Move constant terms to the other side
4. Solve for the variable
5. Flip sign if multiplying/dividing by negative

Examples:

Example 1: Solve \( 4(x - 3) + 5 < 21 \)

Step 1: Distribute: \( 4x - 12 + 5 < 21 \)

Step 2: Combine: \( 4x - 7 < 21 \)

Step 3: Add 7: \( 4x < 28 \)

Step 4: Divide by 4: \( x < 7 \)

Example 2: Solve \( 3x + 7 - x \geq 15 \)

Combine like terms: \( 2x + 7 \geq 15 \)

Subtract 7: \( 2x \geq 8 \)

Divide by 2: \( x \geq 4 \)

Example 3: Solve \( -2(3x + 1) > 16 \)

Distribute: \( -6x - 2 > 16 \)

Add 2: \( -6x > 18 \)

Divide by -6 and flip: \( x < -3 \)

8. Solve Inequalities with Variables on Both Sides

Steps:

1. Simplify both sides if needed
2. Move all variable terms to one side
3. Move all constant terms to the other side
4. Solve for the variable
5. Flip sign if needed

With Integers:

Example 1: Solve \( 5x + 8 > 2x + 17 \)

Subtract \( 2x \): \( 3x + 8 > 17 \)

Subtract 8: \( 3x > 9 \)

Divide by 3: \( x > 3 \)

Example 2: Solve \( 7x - 5 \leq 4x + 10 \)

Subtract \( 4x \): \( 3x - 5 \leq 10 \)

Add 5: \( 3x \leq 15 \)

Divide by 3: \( x \leq 5 \)

With Decimals:

Example 3: Solve \( 2.5x + 3 < 1.5x + 8 \)

Subtract \( 1.5x \): \( 1x + 3 < 8 \)

Subtract 3: \( x < 5 \)

Example 4: Solve \( 0.4x - 2 \geq 0.1x + 1 \)

Subtract \( 0.1x \): \( 0.3x - 2 \geq 1 \)

Add 2: \( 0.3x \geq 3 \)

Divide by 0.3: \( x \geq 10 \)

Tip for Decimals:

You can multiply both sides by a power of 10 to eliminate decimals first!

Example: \( 0.5x + 2 < 0.3x + 5 \)

Multiply by 10: \( 5x + 20 < 3x + 50 \)

Then solve normally

9. Graphing Solutions to Inequalities

Strategy: After solving, graph the solution on a number line.

Complete Process:

1. Solve the inequality algebraically
2. Identify the critical value
3. Determine circle type (open or closed)
4. Determine arrow direction
5. Draw the graph

Examples:

Example 1: Solve and graph \( 2x + 3 > 11 \)

Subtract 3: \( 2x > 8 \)

Divide by 2: \( x > 4 \)

Graph: Open circle at 4, arrow right →

Example 2: Solve and graph \( -3x + 5 \leq 14 \)

Subtract 5: \( -3x \leq 9 \)

Divide by -3 and flip: \( x \geq -3 \)

Graph: Closed circle at -3, arrow right →

Example 3: Solve and graph \( 4x - 7 < x + 5 \)

Subtract \( x \): \( 3x - 7 < 5 \)

Add 7: \( 3x < 12 \)

Divide by 3: \( x < 4 \)

Graph: Open circle at 4, arrow left ←

10. Word Problems with Inequalities

Key Phrases:

Examples:

Example 1: A taxi charges $3 plus $2 per mile. How many miles can you travel if you have at most $25?

Let \( m \) = number of miles

Inequality: \( 3 + 2m \leq 25 \)

Subtract 3: \( 2m \leq 22 \)

Divide by 2: \( m \leq 11 \)

Answer: You can travel at most 11 miles.

Example 2: Sarah wants to score an average of at least 85 on four tests. Her first three scores are 82, 88, and 84. What must she score on the fourth test?

Let \( x \) = fourth test score

Inequality: \( \frac{82 + 88 + 84 + x}{4} \geq 85 \)

Simplify: \( \frac{254 + x}{4} \geq 85 \)

Multiply by 4: \( 254 + x \geq 340 \)

Subtract 254: \( x \geq 86 \)

Answer: Sarah must score at least 86 on the fourth test.

Example 3: A gym membership costs $50 per month plus a one-time fee of $100. How many months can you afford if you have less than $400?

Let \( n \) = number of months

Inequality: \( 100 + 50n < 400 \)

Subtract 100: \( 50n < 300 \)

Divide by 50: \( n < 6 \)

Answer: You can afford less than 6 months (so 5 months maximum).

Quick Reference Guide

💡 Key Tips for Solving Inequalities

• ✓ ALWAYS flip the inequality sign when multiplying or dividing by a negative number
• ✓ Open circle (○) for < or > — value NOT included
• ✓ Closed circle (●) for ≤ or ≥ — value IS included
• ✓ Arrow direction: Left for "less than," Right for "greater than"
• ✓ Check your solution: Pick a test value and verify it works
• ✓ Simplify first: Distribute and combine like terms before solving
• ✓ Show all steps: Especially when flipping the sign
• ✓ "At least" means ≥ (greater than or equal to)
• ✓ "At most" means ≤ (less than or equal to)
• ✓ Remember: Inequalities have infinite solutions, not just one!