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Algebra 1

2.11.2 Finding Solutions to Inequalities on the Coordinate Plane

Algebra 12.11.2 Finding Solutions to Inequalities on the Coordinate Plane

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Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Here are four inequalities. Study each inequality assigned to your group and work together to:

  • Use the graphing tool or technology outside the course to graph each inequality. (The Desmos tool is provided).
  • Find some coordinate pairs that represent solutions to the inequality.
  • Find some coordinate pairs that do not represent solutions.
  • Plot both sets of points. Either use two different colors or two different symbols like X and O.
  • Plot enough points until you start to see the region that contains solutions and the region that contains non-solutions. Look for a pattern describing the region where solutions are plotted.
1.

x y x y

2.

2 y 4 2 y 4

3.

3 x < 0 3 x < 0

4.

x + y > 10 x + y > 10

Video: Finding Solutions to Inequalities on the Coordinate Plane

Watch the following video to learn more about solutions to inequalities on the coordinate plane.

Self Check

Which ordered pair is a solution to the inequality graphed here?


y > 4 5 x 8 5

  1. (2, -2)
  2. (-4, 0)
  3. (4, 0)
  4. (0, -4)

Additional Resources

Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

Linear Inequality

A linear inequality is an inequality that can be written in one of the following forms:

Ax+By>CAx+By>C

Ax+ByCAx+ByC

Ax+By<CAx+By<C

Ax+ByCAx+ByC

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality x>3x>3 is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See the figure below.

A number line with an open parenthesis is graphed at three. A bold arrowed line extending right, shades the numbers greater than 3.

Similarly, linear inequalities in two variables have many solutions. Any ordered pair(x,y)(x,y) that makes an inequality true when we substitute in the values is a solution to a linear inequality.

Solution to a Linear Inequality

An ordered pair(x,y)(x,y) is a solution to a linear inequality if the inequality is true when we substitute the values of xx and yy.

Example

Determine whether each ordered pair is a solution to the inequality y>x+4y>x+4:

  1. (0,0)(0,0)
  2. (1,6)(1,6)
  3. (2,6)(2,6)
  4. (5,15)(5,15)
  5. (8,12)(8,12)

Solution

1. No; Substitute 0 for xx and 0 for yy.

y>x+4y>x+4

0>0+40>0+4

0404

2. Yes; Substitute 1 for xx and 6 for yy.

y>x+4y>x+4

6>1+46>1+4

6>56>5

3. No; Substitute 2 for xx and 6 for yy.

y>x+4y>x+4

6>2+46>2+4

6666

4. No; Substitute -5 for xx and -15 for yy.

y>x+4y>x+4

15>5+415>5+4

15611561

5. Yes; Substitute -8 for xx and 12 for yy.

y>x+4y>x+4

12>8+412>8+4

12>412>4

Try it

Try It: Verify Solutions to an Inequality in Two Variables

Determine whether each ordered pair is a solution to the inequality y>x3y>x3.

1.

( 0 , 0 ) ( 0 , 0 )

2.

( 4 , 9 ) ( 4 , 9 )

3.

( 2 , 1 ) ( 2 , 1 )

4.

( 5 , 3 ) ( 5 , 3 )

5.

( 5 , 1 ) ( 5 , 1 )

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