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By the end of this section, you will be able to:
  • Graph inequalities on the number line
  • Solve inequalities using the Subtraction and Addition Properties of inequality
  • Solve inequalities using the Division and Multiplication Properties of inequality
  • Solve inequalities that require simplification
  • Translate to an inequality and solve

Before you get started, take this readiness quiz.

  1. Translate from algebra to English: 15 > x .
    If you missed this problem, review [link] .
  2. Solve: n 9 = −42 .
    If you missed this problem, review [link] .
  3. Solve: −5 p = −23 .
    If you missed this problem, review [link] .
  4. Solve: 3 a 12 = 7 a 20 .
    If you missed this problem, review [link] .

Graph inequalities on the number line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality x > 3 true? Are you thinking, ‘ x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality x > 3 .

We show the solutions to the inequality x > 3 on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of x > 3 is shown in [link] . Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis.
The inequality x > 3 is graphed on this number line.

The graph of the inequality x 3 is very much like the graph of x > 3 , but now we need to show that 3 is a solution, too. We do that by putting a bracket at x = 3 , as shown in [link] .

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket.
The inequality x 3 is graphed on this number line.

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

Graph on the number line:

x 1 x < 5 x > 1

Solution

  1. x 1
    This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at x = 1 to show that it is included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket.
  2. x < 5
    This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at x = 5 to show it is not included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 5 is graphed on the number line, with an open parenthesis at x equals 5, and a red line extending to the right of the parenthesis.
  3. x > 1
    This means all numbers greater than −1 , but not including −1 . We shade in all the numbers on the number line to the right of −1 , then put a parenthesis at x = −1 to show it is not included.
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 1 is graphed on the number line, with an open parenthesis at x equals negative 1, and a red line extending to the right of the parenthesis.
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Graph on the number line: x 1 x > 2 x < 3


  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the left of the bracket.

  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 2 is graphed on the number line, with an open parenthesis at x equals 2, and a dark line extending to the right of the parenthesis.

  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the left of the parenthesis.
Got questions? Get instant answers now!

Graph on the number line: x > 2 x < 3 x −1


  1. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the right of the parenthesis.

  2. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 3 is graphed on the number line, with an open parenthesis at x equals negative 3, and a dark line extending to the left of the parenthesis.

  3. This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.
Got questions? Get instant answers now!

We can also represent inequalities using interval notation. As we saw above, the inequality x > 3 means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation , we express x > 3 as ( 3 , ) . The symbol is read as ‘infinity’. It is not an actual number. [link] shows both the number line and the interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.
The inequality x > 3 is graphed on this number line and written in interval notation.

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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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