Learn on PengienVision, Algebra 1Chapter 4: Systems of Linear Equations and Inequalities

Lesson 5: Systems of Linear Inequalities

In this Grade 11 enVision Algebra 1 lesson from Chapter 4, students learn how to graph and solve systems of linear inequalities by identifying the overlapping shaded regions that represent all ordered pair solutions. The lesson covers key concepts including boundary lines (solid vs. dashed), parallel systems with no solution, and writing a system of inequalities from a graph. Students also apply these skills to real-world problems, setting up and interpreting systems with constraints such as budget limits and minimum quantity requirements.

Section 1

System of Linear Inequalities

Property

Two or more linear inequalities grouped together form a system of linear inequalities.

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities.

Examples

Determine whether the ordered pair is a solution to the system:

\begin{cases} x + 2y \leq 8 \\ 3x - y > 5 \end{cases}

Is $(2, 3)$ a solution? For the first inequality, $2 + 2(3) \leq 8$ , which is $8 \leq 8$ (True). For the second, $3(2) - 3 > 5$ , which is $3 > 5$ (False). Since one is false, $(2, 3)$ is not a solution.
Is $(4, 1)$ a solution? For the first inequality, $4 + 2(1) \leq 8$ , which is $6 \leq 8$ (True). For the second, $3(4) - 1 > 5$ , which is $11 > 5$ (True). Since both are true, $(4, 1)$ is a solution.
Is $(-1, -10)$ a solution? For the first inequality, $-1 + 2(-10) \leq 8$ , which is $-21 \leq 8$ (True). For the second, $3(-1) - (-10) > 5$ , which is $7 > 5$ (True). Since both are true, $(-1, -10)$ is a solution.

Section 2

Solving a System of Inequalities by Graphing

Property

To solve a system of linear inequalities by graphing, follow a step-by-step layering process:

Graph the boundary line for the first inequality (solid for $\leq, \geq$ ; dashed for $<, >$ ). Shade its solution half-plane lightly.
On the exact same coordinate grid, graph the boundary line for the second inequality. Shade its solution half-plane lightly.
Identify the intersection where the shadings from both inequalities overlap. Make this overlapping region noticeably darker. This dark region is the final solution to the system.

Examples

Example 1: Solve the system $y > x - 2$ and $y \leq -x + 4$ .

Graph a dashed line for $y = x - 2$ and shade above it.
Graph a solid line for $y = -x + 4$ and shade below it.
The final solution is the dark wedge-shaped region where the two shadings overlap.

Example 2: Solve the system $x < 3$ and $y \geq -2$ .

Graph a dashed vertical line at $x = 3$ and shade everything to its left.
Graph a solid horizontal line at $y = -2$ and shade everything above it.
The solution is the top-left rectangular quadrant defined by the crossing of these two lines.

Examples

Solve the system:
$\begin{cases} y > x - 2 \\ y \leq -x + 4 \end{cases}$
Graph a dashed line for $y = x - 2$ and shade above it. Graph a solid line for $y = -x + 4$ and shade below it. The overlapping region is the solution.
Solve the system:
$\begin{cases} x + y < 5 \\ y > 1 \end{cases}$
Graph a dashed line for $x+y=5$ and shade below it. Graph a dashed horizontal line for $y=1$ and shade above it. The solution is the overlapping triangle-like region.
Solve the system:
$\begin{cases} x < 3 \\ y \geq -2 \end{cases}$
Graph a dashed vertical line at $x=3$ and shade to the left. Graph a solid horizontal line at $y=-2$ and shade above. The solution is the top-left quadrant defined by these lines.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

System of Linear Inequalities

Property

Two or more linear inequalities grouped together form a system of linear inequalities.

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities.

Examples

Determine whether the ordered pair is a solution to the system:

\begin{cases} x + 2y \leq 8 \\ 3x - y > 5 \end{cases}

Is $(2, 3)$ a solution? For the first inequality, $2 + 2(3) \leq 8$ , which is $8 \leq 8$ (True). For the second, $3(2) - 3 > 5$ , which is $3 > 5$ (False). Since one is false, $(2, 3)$ is not a solution.
Is $(4, 1)$ a solution? For the first inequality, $4 + 2(1) \leq 8$ , which is $6 \leq 8$ (True). For the second, $3(4) - 1 > 5$ , which is $11 > 5$ (True). Since both are true, $(4, 1)$ is a solution.
Is $(-1, -10)$ a solution? For the first inequality, $-1 + 2(-10) \leq 8$ , which is $-21 \leq 8$ (True). For the second, $3(-1) - (-10) > 5$ , which is $7 > 5$ (True). Since both are true, $(-1, -10)$ is a solution.

Section 2

Solving a System of Inequalities by Graphing

Property

To solve a system of linear inequalities by graphing, follow a step-by-step layering process:

Graph the boundary line for the first inequality (solid for $\leq, \geq$ ; dashed for $<, >$ ). Shade its solution half-plane lightly.
On the exact same coordinate grid, graph the boundary line for the second inequality. Shade its solution half-plane lightly.
Identify the intersection where the shadings from both inequalities overlap. Make this overlapping region noticeably darker. This dark region is the final solution to the system.

Examples

Example 1: Solve the system $y > x - 2$ and $y \leq -x + 4$ .

Graph a dashed line for $y = x - 2$ and shade above it.
Graph a solid line for $y = -x + 4$ and shade below it.
The final solution is the dark wedge-shaped region where the two shadings overlap.

Example 2: Solve the system $x < 3$ and $y \geq -2$ .

Examples

Solve the system:
$\begin{cases} y > x - 2 \\ y \leq -x + 4 \end{cases}$
Graph a dashed line for $y = x - 2$ and shade above it. Graph a solid line for $y = -x + 4$ and shade below it. The overlapping region is the solution.
Solve the system:
$\begin{cases} x + y < 5 \\ y > 1 \end{cases}$
Graph a dashed line for $x+y=5$ and shade below it. Graph a dashed horizontal line for $y=1$ and shade above it. The solution is the overlapping triangle-like region.
Solve the system:
$\begin{cases} x < 3 \\ y \geq -2 \end{cases}$
Graph a dashed vertical line at $x=3$ and shade to the left. Graph a solid horizontal line at $y=-2$ and shade above. The solution is the top-left quadrant defined by these lines.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

System of Linear Inequalities

Property

Two or more linear inequalities grouped together form a system of linear inequalities.

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities.

Examples

Determine whether the ordered pair is a solution to the system:

\begin{cases} x + 2y \leq 8 \\ 3x - y > 5 \end{cases}

Is $(2, 3)$ a solution? For the first inequality, $2 + 2(3) \leq 8$ , which is $8 \leq 8$ (True). For the second, $3(2) - 3 > 5$ , which is $3 > 5$ (False). Since one is false, $(2, 3)$ is not a solution.
Is $(4, 1)$ a solution? For the first inequality, $4 + 2(1) \leq 8$ , which is $6 \leq 8$ (True). For the second, $3(4) - 1 > 5$ , which is $11 > 5$ (True). Since both are true, $(4, 1)$ is a solution.
Is $(-1, -10)$ a solution? For the first inequality, $-1 + 2(-10) \leq 8$ , which is $-21 \leq 8$ (True). For the second, $3(-1) - (-10) > 5$ , which is $7 > 5$ (True). Since both are true, $(-1, -10)$ is a solution.

Section 2

Solving a System of Inequalities by Graphing

Property

To solve a system of linear inequalities by graphing, follow a step-by-step layering process:

Graph the boundary line for the first inequality (solid for $\leq, \geq$ ; dashed for $<, >$ ). Shade its solution half-plane lightly.
On the exact same coordinate grid, graph the boundary line for the second inequality. Shade its solution half-plane lightly.
Identify the intersection where the shadings from both inequalities overlap. Make this overlapping region noticeably darker. This dark region is the final solution to the system.

Examples

Example 1: Solve the system $y > x - 2$ and $y \leq -x + 4$ .

Graph a dashed line for $y = x - 2$ and shade above it.
Graph a solid line for $y = -x + 4$ and shade below it.
The final solution is the dark wedge-shaped region where the two shadings overlap.

Example 2: Solve the system $x < 3$ and $y \geq -2$ .

Examples

Solve the system:
$\begin{cases} y > x - 2 \\ y \leq -x + 4 \end{cases}$
Graph a dashed line for $y = x - 2$ and shade above it. Graph a solid line for $y = -x + 4$ and shade below it. The overlapping region is the solution.
Solve the system:
$\begin{cases} x + y < 5 \\ y > 1 \end{cases}$
Graph a dashed line for $x+y=5$ and shade below it. Graph a dashed horizontal line for $y=1$ and shade above it. The solution is the overlapping triangle-like region.
Solve the system:
$\begin{cases} x < 3 \\ y \geq -2 \end{cases}$
Graph a dashed vertical line at $x=3$ and shade to the left. Graph a solid horizontal line at $y=-2$ and shade above. The solution is the top-left quadrant defined by these lines.