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# Linear Inequalities in Two Variables

Sep 15, 2022

## Key Concepts

• Understand an Inequality in Two Variables.
• Rewrite an inequality to graph it.
• Write an inequality from a graph.
• Inequalities in one variable in the Coordinate plane.

## Introduction

### Linear Inequality

A Linear inequality in two variables is an expression that can be put in the form

ax + by < c

where a, b and c are real numbers (where a and b are not both 0‟s ). The inequality symbol can be any one of the following four:

< , ≤, >, ≥

#### Solution of an inequality

Solution of an inequality is any ordered pair (x, y) that makes the inequality true.

#### Boundary line

It is a line that divides a coordinate plane into two half planes.

#### Half-plane

It is the part of the coordinate plane on one side of a line, which may include the line.

#### Steps to graph an inequality on coordinate plane

1. Rewrite the inequality so that it is in slope-intercept form.

• y = mx + b

2. Plot the y-intercept (b)

3. Use the slope (m) to find other points on the line.

4. Draw the line

• Solid if <= or >=
• Dotted if < or >

5. Shade above or below the line

• Above if > or >=
• Below if < or <=

### Understand an Inequality in Two Variables

Example 1:

What is the solution of the inequality y > 2x -5?

Solution:

Step 1: The equation is already in slope-intercept form. Start by plotting the y-intercept (b = -5)

Step 2: Now use the slope to find other points on the line.

Step 3: Draw a dotted or solid line through the coordinates.

This line will be dotted since the inequality is >

Step 4: Shade above the line to show all of the coordinates that are solutions.

Example 2:

What is the solution of the inequality y ≥ 2x – 4?

Solution:

Step 1: The slope is 2 and the y-intercept is -4.  Use this information to graph the two points needed to draw your line.

y2x – 4 uses the inequality≥, so the line should be solid.  Therefore, draw a solid line through the two points.

Step 2: y2x – 4 uses the inequality≥, so shade above the solid line.

### Rewrite an inequality to Graph it

Example 3:

A school has \$600 to buy molecular sets for students to build models. Write and graph an inequality that represents the number of each type of molecular set the school can buy.

Solution:

Formulate:

Let x = number of large kits

Let y = number of small kits

The total money to buy molecular sets for students is \$600.

24x + 12y ≤ 600

Compute:

Solve the equation for y

24x + 12y ≤ 600

12y ≤ –24+ 600

y ≤ –2+ 50

### Graph the inequality

#### Interpret

Any point in the shaded region or on the boundary line is a solution of the inequality. However, since it is not possible to buy a negative number of large kits or small kits, you must exclude negative values for each.

### Write an inequality from a graph

Example 4:

What inequality does the graph represent?

Solution:

Determine the equation of the boundary line.

The graph is shaded below the boundary line and the boundary line is solid, so the inequality symbol is ≤.

The inequality shown by the graph is y ≤ x + 1.

### Inequalities in one variable in the Coordinate plane

Example 5:

What is the graph of the inequality in the coordinate plane?

A.

y > –2

Solution:

You have graphed the solution of a one-variable inequality on a number line.

Notice that the solution on the number line matches the shaded area for any vertical line on the coordinate grid. This is because x can be any number, and the inequality will still be y > –2.

B.

x ≤ 1

Solution:

You have graphed the solution of a one-variable inequality on a number line.

You can write x ≤ 1 as x + 0 •  y ≤ 1. The inequality is true for all x, whenever x ≤ 1. Imagine stacking copies of the solution on the number line on top of each other, one for each y-value. The combined solutions graphed on the number line make up the shaded region on the coordinate plane.

## Exercise

• Shade ______________ the boundary line for solutions that are less than the inequality.
• Shade ________________ the boundary line for solutions that are greater than the inequality.
• What is the graph of the inequality in the coordinate plane?

x > 5

• What is the graph of the inequality in the coordinate plane?

y < -2

• Describe the graph of the following inequality.

y < –3x + 5

• Describe the graph of the following inequality.

y ≥ –3x + 5

• What inequality does the following graph represents?
• What inequality does the following graph represents?
• Tell whether each ordered pair is a solution of the inequality y > x + 1.
• (0, 1)
• (3, 5)
• A soccer team holds a banquet at the end of the season. The team needs to seat at least 100 people and plans to use two different-sized tables. A small table can seat 6 people, and a large table can seat 8 people. Write a linear inequality that represents the numbers of each size table the team needs. Graph the inequality. If the school has 5 small tables and 9 large tables, will this be enough for the banquet?

### What have we learned

• Understand an Inequality in Two Variables and find the solution.
• Rewrite an inequality from the given scenario and then graph it.
• Read a graph and write an inequality from it.
• Make a coordinate plane for Inequalities in one variable.

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