SAT Math Lesson Plan 4: Systems of Linear Inequalities

Lesson Summary

This 45-minute lesson is part of Media4Math’s 35-lesson SAT Math prep series. It focuses on graphing systems of linear inequalities, a skill found in the Heart of Algebra domain, which accounts for approximately 33% of the SAT Math section. While the number of questions on systems of inequalities varies, being able to interpret and graph them is critical.

Students will learn how to graph individual inequalities, identify boundary lines and shading, and find the solution region where inequalities intersect. This lesson includes multiple examples, self-study tips, a warm-up problem, a review section, and a 10-question multiple-choice quiz. Real-world connections are also explored to reinforce application of these skills in context.

Lesson Objectives

Graph systems of linear inequalities in two variables.
Identify the solution region that satisfies all inequalities in the system.
Apply systems of linear inequalities to real-world scenarios.

Common Core Standards

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 inequalities and their graphs.
Ability to graph linear equations in two variables.
Familiarity with shading regions on the coordinate plane.

Key Vocabulary

System of Linear Inequalities – A set of two or more linear inequalities in the same variables.
Solution Region – The area on the coordinate plane where the shaded regions of all inequalities in the system overlap.
Boundary Line – The line corresponding to the equality 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: Graph the inequality $ y > -\frac{1}{2}x + 3 $.

Steps:

Rewrite in slope-intercept form: Already done.
Slope: -1/2; y-intercept: 3
Use a dashed line for the boundary since it's a strict inequality ( > )
Pick a test point, like (0, 0): $ 0 > -\frac{1}{2}(0) + 3 \rightarrow 0 > 3 $ → false
Shade the region above the line (opposite side of the test point)

Graph it on Desmos: Click here to graph it on Desmos

Final Answer: The solution includes all points above the line $ y = -\frac{1}{2}x + 3 $, but not on the line itself.

Self-Study Tip: Use Desmos to graph several inequalities and observe how the slope and y-intercept affect the shaded region. Practice using test points to verify shading direction.

Teach

When working with a system of linear inequalities, the goal is to identify the region on the coordinate plane that satisfies all inequalities in the system. Each inequality divides the plane into two half-planes. The solution to the system is the intersection of all shaded regions. Follow these key steps for each inequality:

Rewrite in slope-intercept form if needed.
Graph the boundary line (solid for ≤ or ≥, dashed for < or >).
Test a point (usually (0, 0)) to determine which side of the line to shade.
Repeat for the second inequality, and then find the overlapping shaded region.

Example 1: Solution in the First Quadrant

System:

\[ \begin{aligned} y &\leq 2x + 1 \\ y &> -x \end{aligned} \]

Step-by-Step:

Graph $ y = 2x + 1 $ with a solid line (≤ symbol).
Graph $ y = -x $ with a dashed line (> symbol).
Use test point (0, 0):
- $ 0 \leq 2(0) + 1 \Rightarrow 0 \leq 1 $ → true → shade below the first line
- $ 0 > -0 \Rightarrow 0 > 0 $ → false → shade above the second line (opposite of test point)
Shade the region below the first line and above the second line.

Final Answer: The solution region lies between the two lines and includes points where the y-value is less than or equal to $ 2x + 1 $ and greater than $ -x $.

Example 2: No Overlapping Region

System:

\[ \begin{aligned} y &< x + 2 \\ y &> x + 5 \end{aligned} \]

Step-by-Step:

Graph both lines using dashed lines because of the < and > symbols.
Graph $ y = x + 2 $ and $ y = x + 5 $ — these are parallel lines.
Test point (0, 0):
- For $ y < x + 2 $: $ 0 < 2 $ → true → shade below the first line
- For $ y > x + 5 $: $ 0 > 5 $ → false → shade above the second line
The shaded regions do not overlap.

Final Answer: There is no solution region. The system is inconsistent.

Example 3: Boundary Lines with Same Slope

System:

\[ \begin{aligned} y &\geq 2x + 1 \\ y &\leq 2x + 3 \end{aligned} \]

Step-by-Step:

Graph both lines as solid (≥ and ≤).
Lines are parallel, but shading is between them:
Pick test point (0,0):
- $ 0 \geq 1 $ → false → shade above $ y = 2x + 1 $
- $ 0 \leq 3 $ → true → shade below $ y = 2x + 3 $
The overlapping region lies between the two lines.

Final Answer: The solution is the strip between the two parallel boundary lines.

Example 4: One Inequality is Horizontal

System:

\[ \begin{aligned} y &\leq 4 \\ x &> 1 \end{aligned} \]

Step-by-Step:

Graph $ y = 4 $ as a horizontal solid line.
Graph $ x = 1 $ as a vertical dashed line.
Test point (0, 0):
- $ y \leq 4 \Rightarrow 0 \leq 4 $ → true → shade below the horizontal line
- $ x > 1 \Rightarrow 0 > 1 $ → false → shade to the right of the vertical line (opposite side)
The solution region is below y = 4 and to the right of x = 1.

Final Answer: The solution region lies in the top-right quadrant, under the line y = 4 and to the right of x = 1.

Example 5: Real-World Context

Problem: A company makes chairs and tables. Each chair requires 2 hours of labor, and each table requires 4 hours. The company has no more than 40 labor hours available. Also, due to demand, they must produce at least 5 chairs.

Step 1: Define variables

Let c = number of chairs, t = number of tables

Step 2: Write the system

Labor: $ 2c + 4t \leq 40 $
Demand: $ c \geq 5 $

Step 3: Graph the system

First inequality: solid line for $ 2c + 4t = 40 $, rearranged: $ t = -\frac{1}{2}c + 10 $
Second inequality: vertical line $ c = 5 $, solid because of ≥
Shade below the first line and to the right of the second line

Final Answer: The solution region is the set of feasible production plans: at least 5 chairs and a labor total within 40 hours.

Review

Key Takeaways

A system of linear inequalities represents the intersection of two or more shaded regions on the coordinate plane.
Dashed lines represent strict inequalities (< or >); solid lines represent inclusive inequalities (≤ or ≥).
The solution region is where all individual shaded regions overlap.
Always test a point (e.g., (0, 0)) to determine the correct region to shade.
Real-world problems often use inequalities to represent constraints on quantities like time, cost, or resources.

Example 1: Solution Region in Bottom-Left

System:

\[ \begin{aligned} y &\leq -x + 3 \\ y &< x - 2 \end{aligned} \]

Graph both lines: solid for the first (≤), dashed for the second (<).
Test point (0,0):
- $ 0 \leq -0 + 3 \rightarrow 0 \leq 3 $ → true → shade below the first line
- $ 0 < 0 - 2 \rightarrow 0 < -2 $ → false → shade below the second line
Overlap occurs in the region below both lines — the bottom-left area of the graph.

Answer: The solution region lies below both lines in the lower-left quadrant.

Example 2: Parallel Lines with Overlapping Regions

System:

\[ \begin{aligned} y &\geq x \\ y &\leq x + 4 \end{aligned} \]

Graph both lines as solid lines (≥ and ≤).
Shading for $ y \geq x $: above the line
Shading for $ y \leq x + 4 $: below the line
Overlap occurs in the strip between the two lines.

Answer: The solution is the band between the lines $ y = x $ and $ y = x + 4 $.

Example 3: Real-World Practice

Problem: A student is choosing between two gym membership plans. Plan A charges \$30 per month plus \$5 per visit. Plan B charges \$50 per month with unlimited visits. The student wants to know for how many visits Plan A is cheaper.

Step 1: Let v be the number of visits

Plan A cost: $ C_A = 30 + 5v $
Plan B cost: $ C_B = 50 $

Step 2: Set up inequality:

$ 30 + 5v < 50 $
$ 5v < 20 \Rightarrow v < 4 $

Step 3: Interpret

If the student makes fewer than 4 visits, Plan A is cheaper.

Answer: Plan A is cheaper for 3 or fewer visits in a month.

Multimedia Resources

To explore the concepts in this lesson further, visit the Multimedia Resources for Lesson 4 page. You’ll find videos, tutorials, and practice activities aligned with this lesson topic.

Quiz

What does the graph of $ y < 2x + 1 $ look like?
A) Solid line, shaded above
B) Solid line, shaded below
C) Dashed line, shaded above
D) Dashed line, shaded below
Which point is in the solution region of this system?
$ y \leq x + 3 $
$ y > -x $
A) (0, 0)
B) (2, 1)
C) (-2, -4)
D) (1, 4)
Which system has no solution?
A) $ y > x $ and $ y < x + 2 $
B) $ y < x + 1 $ and $ y > x + 3 $
C) $ y \geq -x $ and $ y \leq x $
D) $ y < 2x $ and $ y < -2x $
Which inequality has a solution region above the boundary line?
A) $ y < -x + 2 $
B) $ y \geq 3x - 1 $
C) $ y \leq x - 4 $
D) $ y < 0.5x + 1 $
What does the solution to a system of inequalities represent?
A) A single point
B) The area between two lines
C) The overlapping region of all shaded areas
D) The intersection point of two boundary lines
The solution to $ y \leq x + 2 $ and $ y \geq x - 1 $ is:
A) Above both lines
B) Below both lines
C) Between the two lines
D) Left of both lines
Which boundary line is dashed?
A) $ y \geq 2x $
B) $ y = 3x $
C) $ y < -x + 5 $
D) $ y \leq -2x - 1 $
Which point is in the solution set of $ y > 2x - 3 $?
A) (0, -4)
B) (1, 0)
C) (2, 1)
D) (-1, -6)
In a real-world problem, if $ 2x + 3y \leq 60 $ represents a budget constraint, what does the inequality represent?
A) Spend more than \$60
B) Spend less than or equal to \$60
C) Spend exactly \$60
D) Spend no money
What is the solution region for $ x \geq 0 $ and $ y \geq 0 $?
A) First quadrant
B) Second quadrant
C) Below the x-axis
D) Left of the y-axis

Answer Key

$ y < 2x + 1 $ uses a dashed line and shades below.
Answer: D
(2, 1):
$ 1 \leq 2 + 3 $ → true, $ 1 > -2 $ → true.
Answer: B
$ y < x + 1 $ and $ y > x + 3 $ have no overlap.
Answer: B
$ y \geq 3x - 1 $ shades above the line.
Answer: B
The solution is the overlapping region of all inequalities.
Answer: C
Between the lines: shaded above $ x - 1 $ and below $ x + 2 $
Answer: C
Dashed lines are used for strict inequalities (< or >)
Answer: C
Test: $ 0 > 2(1) - 3 = -1 $? → $ 0 > -1 $ → true
Answer: B
Budget constraint: less than or equal to \$60
Answer: B
$ x \geq 0 $ and $ y \geq 0 $ is the first quadrant
Answer: A