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By the end of this section, you will be able to:
  • Verify solutions to an inequality in two variables
  • Recognize the relation between the solutions of an inequality and its graph
  • Graph linear inequalities

Before you get started, take this readiness quiz.

  1. Solve: 4 x + 3 > 23 .
    If you missed this problem, review [link] .
  2. Translate from algebra to English: x < 5 .
    If you missed this problem, review [link] .
  3. Evaluate 3 x 2 y when x = 1 , y = −2 .
    If you missed this problem, review [link] .

Verify solutions to an inequality in two variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. 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 would make a profit.

Linear inequality

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

A x + B y > C A x + B y C A x + B y < C A x + B y C

where A and B are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality x > 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 [link] .

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.

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

Solution of a linear inequality

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

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

( 0 , 0 ) ( 1 , 6 ) ( 2 , 6 ) ( −5 , −15 ) ( −8 , 12 )


  1. ( 0 , 0 ) .
    . .
    Simplify. .
    So, ( 0 , 0 ) is not a solution to y > x + 4 .

  2. ( 1 , 6 ) .
    . .
    Simplify. .
    So, ( 1 , 6 ) is a solution to y > x + 4 .

  3. ( 2 , 6 ) .
    . .
    Simplify. .
    So, ( 2 , 6 ) is not a solution to y > x + 4 .

  4. ( −5 , −15 ) .
    . .
    Simplify. .
    So, ( −5 , −15 ) is not a solution to y > x + 4 .

  5. ( −8 , 12 ) .
    . .
    Simplify. .
    So, ( −8 , 12 ) is a solution to y > x + 4 .
Got questions? Get instant answers now!
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Determine whether each ordered pair is a solution to the inequality y > x 3 :

( 0 , 0 ) ( 4 , 9 ) ( −2 , 1 ) ( −5 , −3 ) ( 5 , 1 )

yes  yes  yes  yes  no

Got questions? Get instant answers now!

Determine whether each ordered pair is a solution to the inequality y < x + 1 :

( 0 , 0 ) ( 8 , 6 ) ( −2 , −1 ) ( 3 , 4 ) ( −1 , −4 )

yes  yes  no  no  yes

Got questions? Get instant answers now!

Recognize the relation between the solutions of an inequality and its graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in [link] again. The point x = 3 separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See [link] .

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”

The solution to x > 3 is the shaded part of the number line to the right of x = 3 .

Similarly, the line y = x + 4 separates the plane into two regions. On one side of the line are points with y < x + 4 . On the other side of the line are the points with y > x + 4 . We call the line y = x + 4 a boundary line.

Boundary line

The line with equation A x + B y = C is the boundary line    that separates the region where A x + B y > C from the region where A x + B y < C .

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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