Math Skill: Graphs to Linear Inequalities

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:

- Find the equation of the boundary line.
- Choose a point (
*x*,*y*) on the shaded side of the line. - Plug
*x*and*y*into the bounday line equation to determine the inequality sign. - If the boundary line is solid, then the inequality sign is either ≥ or ≤.
- 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:Plugxandyinto 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 = -5

4x + 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:Plugxandyinto the bounday line equation to determine the inequality sign.

y?

-5

4x+ 30 ?

-5

4(5) + 3 Plug in (5, 0) 0 >

-13

4"Greater than" to make the inequality true y≥

-5

4x+ 3Boundary 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: |

## Answers

### Complexity=1, Mode=simple

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

# | Problem | Correct Answer | Your Answer |
---|---|---|---|

1 | Inequality: | ||

SolutionVertical 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 |

# | Problem | Correct Answer | Your Answer |
---|---|---|---|

2 | Inequality: | ||

SolutionHorizontal 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**

# | Problem | Correct Answer | Your Answer | |
---|---|---|---|---|

1 | Inequality: | |||

SolutionThe 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 |

# | Problem | Correct Answer | Your Answer | |
---|---|---|---|---|

2 | Inequality: | |||

SolutionThe 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**

# | Problem | Correct Answer | Your Answer | |
---|---|---|---|---|

1 | Inequality: | |||

SolutionThe 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 |

# | Problem | Correct Answer | Your Answer | |
---|---|---|---|---|

2 | Inequality: | |||

SolutionThe slope is ^{3}/_{5}The y intercept is 4 The shaded region is above the line, so we use a greater than sign. The line is dashed, so use > for greater than. Plugging this into y=mx+b results in |