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Solving Two-Step Linear Inequalities
(single variable)
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Click Basic Inequalities or Solving One Step Linear Inequalities for introductory information.

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If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one important exception ...

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... when you multiply (or divide) an inequality by a negative value,
you must change the direction of the inequality.

Let's see why this "exception" is actually needed.

We know that 3 is less than 7.
Now, lets multiply both sides by -1.
Examine the results (the products).
 ... written 3 < 7.
... written (-1)(3) ? (-1)(7)
... written -3 ? -7

On a number line, -3 is to the right of -7, making -3 greater than -7.
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-3 > -7
We have to reverse the direction of the inequality, when we multiply or divide by a negative value, in order to maintain a "true" statement.

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ex1

Solve and graph the solution set of:   4x < 24
Proceed as you would when solving a linear equation such as 4x = 24:
Divide both sides by 4.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 6 (since x can not equal 6) and an arrow to the left (since our symbol is less than).

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ex2

Solve and graph the solution set of:   -5x greaterequala 25
Divide both sides by -5.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for -5 (since x can equal -5) and an arrow to the left (since our final symbol is less than or equal to).

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ex3

Solve and graph the solution set of:   3x + 4 > 13
Proceed as you would when solving a linear equation such as 3x + 4 = 13:
Subtract 4 from both sides.
Divide both sides by 3.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 3 (since x can not equal 3) and an arrow to the right (since our symbol is greater than).

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ex7

Solve and graph the solution set of:   9 - 2x lessequal 3
Subtract 9 from both sides.
Divide both sides by -2.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for 3 (since x can equal 3) and an arrow to the right (since our symbol is greater than or equal to).

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ex5

Solve and graph the solution set of:   3 - 4x + 2 lessequal 13
Combine 3 and 2 on the left side.
Subtract 5 from both sides.
Divide both sides by -4.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for -2 (since x can equal -2).

 

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ex6

Solve and graph the solution set of:   10 < 3(2x + 4) - 4x
Distribute across parentheses.
Combine the x-values.
Subtract 12 from both sides.
Divide both sides by 2.

Note: There was no multiplication or division by a negative value, so the inequality symbol did not get reversed.

Remember: -1 < x can be reversed so x is on the left side, and written as x > -1.

Graph using a closed circle for -2 (since x can equal -2).

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hint gal
Yes, there is a way to determine solutions for inequalities on your graphing calculator. Click the calculator at the right to see how to use the calculator with single variable inequalities.
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For calculator help with inequalities
(single variable)
click here.



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