Math Skill: Graphs to Linear Inequalities
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Graphs to Linear Inequalities

For this topic, you need to know how to find the equation of a line.
To review the Graphs to Linear Equations topic, see here.

In this topic, you will find the inequality based on its graph.

To find the ineqaulity, follow these steps:

1. Find the equation of the boundary line.
2. Choose a point (x, y) on the shaded side of the line.
3. Plug x and y into the bounday line equation to determine the inequality sign.
1. If the boundary line is solid, then the inequality sign is either ≥ or ≤.
2. If the boundary line is dotted, then the inequality sign is either > or <.

Example 1:

Write the inequality that represents this graph. Examples: y>3x/4, x<=2, y>=1, y>-2x
 Inequality:
Step 1: Find the equation of the boundary line.
The boundary line is vertical. Its equation is x = -1.

Step 2: Choose a point (x, y) on the shaded side of the line.
Let's pick (-3, 0) on the shaded side.

Step 3: Plug x and y into the bounday line equation to determine the inequality sign.

 x ? -1 -3 ? -1 Plug in (-3,0) -3 < -1 "Less than" makes the inequality true x < -1 Boundary line is dashed

Example 2:

Write the inequality in y=mx+b format. Examples: y>2x-3, y<=-4x/3+1
 Inequality:
Step 1: Find the equation of the boundary line.
The slope of the line is -5/4 and the y-intercept is 3.
 So the equation of the boundary line is y = -54 x + 3.

Step 2: Choose a point (x, y) on the shaded side of the line.
Let's pick (5, 0) on the shaded side.

Step 3: Plug x and y into the bounday line equation to determine the inequality sign.

y ?
 -54 x + 3
0 ?
 -54 (5) + 3
Plug in (5, 0)
0 >
 -134
"Greater than" to make the inequality true
y
 -54 x + 3
Boundary line is solid

Complexity=1, Mode=simple

Write the inequality that represents this graph. Examples: y>3x/4, x<=2, y>=1, y>-2x
 1.   Inequality: 2.   Inequality:

Complexity=1, Mode=y=mx+b

Write the inequality in y=mx+b format. Examples: y>2x-3, y<=-4x/3+1
 1.   Inequality: 2.   Inequality:

Complexity=2, Mode=y=mx+b

Write the inequality in y=mx+b format. Examples: y>2x-3, y<=-4x/3+1
 1.   Inequality: 2.   Inequality:

Complexity=1, Mode=simple

Write the inequality that represents this graph. Examples: y>3x/4, x<=2, y>=1, y>-2x
1

Inequality:
Solution
Vertical lines have undefined slope, and are expressed in the form x=[x-intercept].
The shaded region is to the right of the line, so we use a greater than sign.
The line is dashed, so use > for greater than.
x > - 3
2

Inequality:
Solution
Horizontal lines have a slope of 0, and are expressed in the form y=[y-intercept]
The shaded region is above the line, so we use a greater than sign.
The line is dashed, so use > for greater than.
y > - 4

Complexity=1, Mode=y=mx+b

Write the inequality in y=mx+b format. Examples: y>2x-3, y<=-4x/3+1
1

Inequality:
Solution
The slope is -1/4
The y intercept is -5
The shaded region is below the line, so we use a less than sign.
The line is solid, so use <= for less than or equal to.
Plugging this into y=mx+b results in

2

Inequality:
Solution
The slope is 1/5
The y intercept is 5
The shaded region is below the line, so we use a less than sign.
The line is solid, so use <= for less than or equal to.
Plugging this into y=mx+b results in

Complexity=2, Mode=y=mx+b

Write the inequality in y=mx+b format. Examples: y>2x-3, y<=-4x/3+1
1

Inequality:
Solution
The slope is 4/5
The y intercept is 1
The shaded region is below the line, so we use a less than sign.
The line is solid, so use <= for less than or equal to.
Plugging this into y=mx+b results in