## 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* ≥ 2*x* – 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.

*y*≥**2***x* **– 4** uses the inequality≥,** **so the line should be solid. Therefore, draw a **solid** line through the two points.

Step 2: *y*≥**2***x* **– 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.

24*x *+ 12*y *≤ 600

**Compute:**

Solve the equation for *y*.

24*x *+ 12*y *≤ 600

12*y *≤ –24*x *+ 600

*y *≤ –2*x *+ 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

Answer:

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

*y *< -2

Answer:

- Describe the graph of the following inequality.

*y* < –3*x* + 5

- Describe the graph of the following inequality.

*y* ≥ –3*x* + 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?

### Concept Map

### 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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