SAT Math Lesson Plan 2: Linear Inequalities

Lesson Summary

Questions involving linear inequalities are a key component of the SAT Math section, falling under the Heart of Algebra category, which accounts for approximately 35% of the test. The Heart of Algebra domain contributes around 13–15 questions to the SAT Math section, though the College Board does not specify the exact proportion dedicated solely to linear inequalities.

Understanding linear inequalities is crucial, as they often appear in real-world contexts and graphical representations. This lesson provides clear, step-by-step instructions for solving and graphing linear inequalities, making it suitable for teachers, tutors, and independent students looking to strengthen their SAT math skills.

Lesson Objectives

• Solve linear inequalities in one variable.
• Graph linear inequalities in two variables on the coordinate plane.
• Interpret and apply linear inequalities in real-world scenarios.

Common Core Standards

• CCSS.MATH.CONTENT.HSA.REI.B.3 – Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
• CCSS.MATH.CONTENT.HSA.REI.D.12 – Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Prerequisite Skills

• Understanding of linear equations and their graphs.
• Ability to solve linear equations algebraically.
• Familiarity with plotting points and graphing lines on the coordinate plane.

Key Vocabulary

• Linear Inequality – An inequality that involves a linear expression in one or two variables, such as \(2x + 3 \leq 7\).
• Solution Set – The set of all values that satisfy the inequality.
• Inequality Symbols:
  • <: Less than
  • >: Greater than
  • ≤: Less than or equal to
  • ≥: Greater than or equal to
• Boundary Line – The line that represents the equation part of a linear inequality. It divides the coordinate plane into regions and helps determine which region satisfies the inequality.

Warm-Up

Goal: Activate prior knowledge with a quick review.

Problem: Solve the following inequality for x:

\[ 3x - 4 \leq 11 \]

Step 1: Isolate the variable

• Add 4 to both sides:
• \( 3x - 4 + 4 \leq 11 + 4 \)
• \( 3x \leq 15 \)

Step 2: Solve for x

• Divide by 3:
• \( x \leq \frac{15}{3} \)
• \( x \leq 5 \)

Step 3: Graph the Solution

• Plot 5 on a number line.
• Since the inequality is "less than or equal to," use a closed circle.
• Shade the region to the left of 5.

Final Answer: The solution set includes all values of x less than or equal to 5.

Teach

Linear inequalities function similarly to linear equations but introduce inequality symbols (<, >, ≤, ≥). When solving inequalities, many of the same algebraic principles apply, but special attention is required when multiplying or dividing by negative numbers, as this reverses the inequality sign.

We will cover three key concepts:

• Solving linear inequalities in one variable.
• Graphing linear inequalities in two variables.
• Understanding how inequalities represent real-world constraints.

Example 1: Solving a One-Variable Inequality

Problem: Solve for x in the inequality:

\[ 5x - 7 \geq 18 \]

Step 1: Isolate the Variable

• Add 7 to both sides:
• \( 5x - 7 + 7 \geq 18 + 7 \)
• \( 5x \geq 25 \)

Step 2: Solve for x

• Divide both sides by 5:
• \( x \geq \frac{25}{5} \)
• \( x \geq 5 \)

Step 3: Graph the Solution

• Plot 5 on a number line.
• Since the inequality is greater than or equal to, use a closed circle on 5.
• Shade the region to the right to indicate all values greater than or equal to 5.

Final Answer: The solution set is all values of x greater than or equal to 5.

Example 2: Solving an Inequality with Division by a Negative Number

Problem: Solve for x in the inequality:

\[ -2x + 4 < 10 \]

Step 1: Isolate the Variable

• Subtract 4 from both sides:
• \( -2x + 4 - 4 < 10 - 4 \)
• \( -2x < 6 \)

Step 2: Divide by the Coefficient

• Divide both sides by -2, remembering to reverse the inequality sign:
• \( x > \frac{6}{-2} \)
• \( x > -3 \)

Step 3: Graph the Solution

• Plot -3 on a number line.
• Since the inequality is greater than, use an open circle on -3.
• Shade the region to the right to indicate all values greater than -3.

Final Answer: The solution set is all values of x greater than -3.

Example 3: Graphing a Linear Inequality in Two Variables

Problem: Graph the inequality:

\[ y \leq 2x - 1 \]

Step 1: Graph the Boundary Line

• Rewrite the inequality as an equation: \( y = 2x - 1 \).
• Identify the slope and y-intercept:
  • Slope: \( m = 2 \) (rise 2, run 1).
  • Y-intercept: \( b = -1 \) (crosses the y-axis at (0,-1)).
• Graph the line y = 2x - 1.

Step 2: Determine Line Style

• Since the inequality is ≤, the boundary line is solid.

Step 3: Shade the Correct Region

• Pick a test point not on the line, such as (0,0).
• Substituting into the inequality:
• \( 0 \leq 2(0) - 1 \) → \( 0 \leq -1 \) (False).
• Since the test point does not satisfy the inequality, shade the region below the line.

Final Answer: The solution set includes all points below the solid line \( y = 2x - 1 \).

Example 4: Graphing a Different Linear Inequality

Problem: Graph the inequality:

\[ y > -\frac{1}{2}x + 3 \]

Step 1: Graph the Boundary Line

• Rewrite as \( y = -\frac{1}{2}x + 3 \).
• Slope: \( -\frac{1}{2} \), Y-intercept: (0,3).

Step 2: Determine Line Style

• Since the inequality is "greater than," the boundary line is dashed.

Step 3: Shade the Correct Region

• Pick a test point such as (0,0).
• \( 0 > -\frac{1}{2}(0) + 3 \), which is false.
• Since the test point does not satisfy the inequality, shade the region above the line.

Example 5: Real-World Application of Linear Inequalities

Problem: A student has no more than $50 to spend on school supplies. Notebooks cost $3 each and pens cost $2 each. Write and graph an inequality to represent the number of notebooks (n) and pens (p) the student can purchase.

Step 1: Define Variables

• Let n = number of notebooks
• Let p = number of pens

Step 2: Write the Inequality

• Each notebook costs $3 → total cost for notebooks = \( 3n \)
• Each pen costs $2 → total cost for pens = \( 2p \)
• Total cost must be ≤ $50:
• \( 3n + 2p \leq 50 \)

Step 3: Interpret the Inequality

• This inequality represents the budget constraint.
• Any combination of notebooks and pens that satisfies \( 3n + 2p \leq 50 \) is affordable for the student.

Step 4: Graph the Inequality (Optional for SAT-style problems)

• To graph, rearrange as \( 2p \leq -3n + 50 \), then divide by 2: \( p \leq -\frac{3}{2}n + 25 \)
• This would be graphed in the first quadrant, since negative values of n or p aren't realistic in this context.

Final Answer: The inequality \( 3n + 2p \leq 50 \) models the student’s budget constraint for purchasing notebooks and pens.

Review

Key Takeaways

• Linear inequalities are similar to linear equations but use inequality symbols (<, >, ≤, ≥).
• Solving inequalities follows the same steps as solving equations, except when multiplying or dividing by a negative number, in which case the inequality sign must be reversed.
• A one-variable inequality represents a range of values on a number line.
• A two-variable inequality represents a region on a coordinate plane, with a boundary line that is either solid (for ≤ or ≥) or dashed (for < or >).
• The solution region for a linear inequality is determined by testing a point in one of the regions separated by the boundary line.

Example 1: Solving a One-Variable Inequality

Problem: Solve for x:

\[ 3x - 5 < 10 \]

Step 1: Add 5 to both sides

• \( 3x - 5 + 5 < 10 + 5 \)
• \( 3x < 15 \)

Step 2: Divide by 3

• \( x < \frac{15}{3} \)
• \( x < 5 \)

Final Answer: x is any number less than 5.

Example 2: Graphing a Two-Variable Inequality

Problem: Graph the inequality:

\[ y > -\frac{2}{3}x + 4 \]

Step 1: Graph the Boundary Line

• Convert to the equation \( y = -\frac{2}{3}x + 4 \).
• Slope: -2/3, y-intercept: (0,4).

Step 2: Determine Line Style

• Since the inequality is "greater than," use a dashed line.

Step 3: Shade the Correct Region

• Pick (0,0) as a test point: \( 0 > -\frac{2}{3}(0) + 4 \), which is false.
• Shade the region above the line.

Multimedia Resources

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Quiz

1. Solve for x: \( 4x - 6 \geq 10 \)
   A) \( x \geq 2 \) 
   B) \( x \geq 4 \) 
   C) \( x \leq 2 \) 
   D) \( x \leq 4 \) 
    
2. Solve for x: \( -5x + 3 > 18 \)
   A) \( x < -3 \) 
   B) \( x > -3 \) 
   C) \( x > 3 \) 
   D) \( x < 3 \) 
    
3. Which of the following inequalities represents a boundary line with a dashed line?
   A) \( y \geq 3x - 5 \) 
   B) \( y \leq -2x + 4 \) 
   C) \( y > \frac{1}{2}x + 2 \) 
   D) \( y \geq -x + 1 \) 
    
4. Which test point should be used to determine shading for the inequality \( y < 2x + 3 \)?
   A) (0,0) 
   B) (1,1) 
   C) (-1,2) 
   D) (3,5) 
    
5. Graph the inequality \( y \leq -\frac{3}{4}x + 5 \). Which statement is true?
   A) The boundary line is solid, and the shading is above the line. 
   B) The boundary line is dashed, and the shading is below the line. 
   C) The boundary line is solid, and the shading is below the line. 
   D) The boundary line is dashed, and the shading is above the line. 
    
6. Solve for x: \( 2x - 8 \leq 12 \)
   A) \( x \geq 5 \) 
   B) \( x \leq 10 \) 
   C) \( x \leq 6 \) 
   D) \( x \geq 2 \) 
    
7. Which inequality represents the solution for \( -3x + 2 \geq 11 \)?
   A) \( x \geq -3 \) 
   B) \( x \leq -3 \) 
   C) \( x \geq 3 \) 
   D) \( x \leq 3 \) 
    
8. Which inequality describes a line with a slope of 4 and a y-intercept of -2?
   A) \( y > 4x - 2 \) 
   B) \( y \leq 4x + 2 \) 
   C) \( y \geq 4x - 2 \) 
   D) \( y < 4x + 2 \) 
    
9. If the solution to an inequality is \( x < 7 \), what type of circle is used on a number line?
   A) Open circle on 7 
   B) Closed circle on 7 
   C) Open circle on -7 
   D) Closed circle on -7 
    
10. Which inequality has a solution region shaded above the boundary line?
    A) \( y < -x + 4 \) 
    B) \( y \leq 2x - 5 \) 
    C) \( y > \frac{1}{3}x + 1 \) 
    D) \( y \leq -\frac{2}{5}x + 3 \) 
     

Answer Key

1. Solution: \( 4x - 6 \geq 10 \) 
   Add 6: \( 4x \geq 16 \) 
   Divide by 4: \( x \geq 4 \) 
   Correct Answer: B
2. Solution: \( -5x + 3 > 18 \) 
   Subtract 3: \( -5x > 15 \) 
   Divide by -5 (flip the sign): \( x < -3 \) 
   Correct Answer: A
3. Explanation: A dashed boundary is used for inequalities with "<" or ">". 
   Correct Answer: C
4. Explanation: (0,0) is the easiest test point, as long as it is not on the boundary line. 
   Correct Answer: A
5. Explanation: Since the inequality uses "≤", the boundary line is solid and shading is below. 
   Correct Answer: C
6. Solution: \( 2x - 8 \leq 12 \) 
   Add 8: \( 2x \leq 20 \) 
   Divide by 2: \( x \leq 10 \) 
   Correct Answer: B
7. Solution: \( -3x + 2 \geq 11 \) 
   Subtract 2: \( -3x \geq 9 \) 
   Divide by -3 (flip sign): \( x \leq -3 \) 
   Correct Answer: B
8. Explanation: The equation has a slope of 4 and a y-intercept of -2. 
   Correct Answer: C
9. Explanation: "Less than" inequalities use an open circle on the boundary. 
   Correct Answer: A
10. Explanation: The solution region is shaded above when the inequality symbol is ">". 
    Correct Answer: C