Linear Inequalities in Two Variables

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. 

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

Step 2: y≥2x – 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 ≤ –24x  + 600 

y ≤ –2x  + 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 < –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?

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.