Graphing Two-Variable Inequalities:

To graph a linear inequality in two variables (x and y), we graph the boundary line and then shade the appropriate region.

Steps to Graph Linear Inequalities:

1. Step 1: Convert to Slope-Intercept Form
   • Rearrange the inequality so y is isolated on the left side
   • Form: \( y < mx + b \) or \( y > mx + b \) or \( y \leq mx + b \) or \( y \geq mx + b \)
2. Step 2: Graph the Boundary Line
   • Solid line (—): Use for \( \leq \) or \( \geq \) (includes the line)
   • Dashed line (- - -): Use for \( < \) or \( > \) (does not include the line)
   • Graph \( y = mx + b \) using slope-intercept method
3. Step 3: Shade the Correct Region
   • Shade above the line: For \( y > mx + b \) or \( y \geq mx + b \)
   • Shade below the line: For \( y < mx + b \) or \( y \leq mx + b \)
   • Test Point Method: Use (0,0) if it's not on the line to verify shading

Quick Memory Tips:

• Greater (>): Shade Above the line
• Less (<): Shade Below the line
• Equal to (≤, ≥): Use Solid line
• Not equal (<, >): Use Dashed line

Example: Graph \( y \leq 2x - 3 \)

Step 1: Already in slope-intercept form

Step 2: Graph \( y = 2x - 3 \) using a solid line (because of ≤)

• y-intercept: -3 (point: 0, -3)

• Slope: 2 (rise 2, run 1)

Step 3: Shade below the line (because y is less than or equal to)

Test: Point (0, 0): \( 0 \leq 2(0) - 3 \) → \( 0 \leq -3 \) (False, so shade opposite side)

How to Write an Inequality from a Graph:

Given a graph with a boundary line and shaded region, you need to write the corresponding inequality.

Steps to Write Inequalities from Graphs:

1. Step 1: Find the Equation of the Boundary Line
   • Identify the y-intercept (b) where the line crosses the y-axis
   • Calculate the slope (m) using \( m = \frac{\text{rise}}{\text{run}} \) or \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
   • Write the equation: \( y = mx + b \)
2. Step 2: Determine the Inequality Symbol
   • Solid line: Use \( \leq \) or \( \geq \)
   • Dashed line: Use \( < \) or \( > \)
   • Shaded above: Use \( > \) or \( \geq \)
   • Shaded below: Use \( < \) or \( \leq \)
3. Step 3: Write the Complete Inequality
   • Replace the = sign in your equation with the correct inequality symbol
   • Verify by testing a point from the shaded region

Decision Chart for Inequality Symbols:

Example: Write the Inequality

Given: A dashed line passing through (0, 1) and (2, 5), with shading above the line.

Step 1: Find the equation

• y-intercept: \( b = 1 \)

• Slope: \( m = \frac{5-1}{2-0} = \frac{4}{2} = 2 \)

• Equation: \( y = 2x + 1 \)

Step 2: Determine symbol

• Dashed line + Shaded above = \( > \)

Answer: \( y > 2x + 1 \)

Solving One-Variable Inequalities:

Solving linear inequalities is similar to solving linear equations, with one critical difference: when you multiply or divide by a negative number, you must reverse the inequality sign.

Steps to Solve Linear Inequalities:

1. Step 1: Simplify Both Sides
   • Remove parentheses using distributive property
   • Combine like terms
   • Clear fractions by multiplying by LCD
2. Step 2: Isolate the Variable
   • Move variable terms to one side
   • Move constant terms to the other side
3. Step 3: Solve for the Variable
   • Divide or multiply to isolate the variable
   • CRITICAL: If multiplying or dividing by a negative number, flip the inequality sign
4. Step 4: Write the Solution
   • Express as an inequality: \( x < a \) or \( x \geq b \)
   • Or use interval notation: \( (-\infty, a) \) or \( [b, \infty) \)

⚠️ CRITICAL RULE - Must Remember!

When multiplying or dividing both sides by a negative number,
you MUST reverse (flip) the inequality symbol!

Rules That DON'T Change the Inequality Sign:

• Addition: \( x + a < b \) → Add or subtract same number to both sides
• Subtraction: \( x - a > b \) → Add or subtract same number to both sides
• Multiplication by Positive: \( \frac{x}{3} \leq b \) → Multiply both sides by positive number
• Division by Positive: \( 3x \geq b \) → Divide both sides by positive number

Example 1: Simple Inequality

Solve: \( 3x - 5 > 10 \)

Step 1: Add 5 to both sides → \( 3x > 15 \)

Step 2: Divide by 3 (positive, so sign stays) → \( x > 5 \)

Solution: \( x > 5 \) or \( (5, \infty) \)

Example 2: Negative Coefficient (FLIP THE SIGN!)

Solve: \( -4x + 7 \leq 19 \)

Step 1: Subtract 7 from both sides → \( -4x \leq 12 \)

Step 2: Divide by -4 (negative, so FLIP ≤ to ≥) → \( x \geq -3 \)

Solution: \( x \geq -3 \) or \( [-3, \infty) \)

Example 3: Variables 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 → \( x < 4 \)

Solution: \( x < 4 \) or \( (-\infty, 4) \)

Graphing Solutions on a Number Line:

After solving a one-variable inequality, we graph the solution set on a number line to visualize all values that satisfy the inequality.

Steps to Graph on a Number Line:

1. Step 1: Draw a Number Line
   • Draw a horizontal line and mark appropriate numbers
   • Include the critical value (boundary point) from your solution
2. Step 2: Mark the Boundary Point
   • Closed circle (●): Use for \( \leq \) or \( \geq \) (includes the value)
   • Open circle (○): Use for \( < \) or \( > \) (does not include the value)
3. Step 3: Shade the Solution Region
   • Shade to the right (→): For \( x > a \) or \( x \geq a \)
   • Shade to the left (←): For \( x < a \) or \( x \leq a \)
   • Use an arrow to show the shading continues infinitely

Visual Guide for Number Line Graphing:

Interval Notation Reference:

Note: Always use ( ) with ∞ and -∞ because infinity is not a specific number

Example 1: Graph \( x \geq -2 \)

Number Line:

Closed circle at -2 (includes -2)

Shade to the right (all values greater than -2)

Interval Notation: \( [-2, \infty) \)

Example 2: Graph \( x < 3 \)

Number Line:

Open circle at 3 (does not include 3)

Shade to the left (all values less than 3)

Interval Notation: \( (-\infty, 3) \)

Example 3: Solve and Graph \( -3x + 5 > 11 \)

Solve:

Subtract 5: \( -3x > 6 \)

Divide by -3 (FLIP!): \( x < -2 \)

Number Line:

Interval Notation: \( (-\infty, -2) \)

• Graph Inequalities (2 variables): Convert to y = form → Use solid/dashed line → Shade above/below
• Write from Graphs: Find line equation → Determine symbol from line type and shading
• Solve Inequalities (1 variable): Solve like equations BUT flip sign when dividing/multiplying by negative
• Graph Solutions (Number line): Use closed/open circle → Shade left/right → Write interval notation