Solve and Graph Linear Inequalities
With an equation such as x = 4, there is a specific value for the
variable. With inequalities, there will be a range of values for
the variable. The inequality symbols are:
> greater than
> greater than or equal to
< less than
< less than or equal to
The inequality is solved exactly the same way as in equality.
Once solved, because there isn’t one specific value for the
variable, it is often useful to indicate the solution on a number
line. The corresponding symbols used on the number line are:
( used for > (greater than)
[ used for > (great than or equal to)
) used for < (less than)
] used for < (less than or equal to)
And then once the graph is drawn, the solution can easily be
converted to interval notation. Interval notation is given as an
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
ordered pair using parentheses and/or brackets, with the first
value being the smaller value and the second value being the
larger value. For greater than (greater than or equal to), the
larger value is ∞, the symbol for infinity. For less than (less
than or equal to), the smaller value will be -∞, the symbol for
negative infinity.
Example 1: Graph the inequality and write the interval notation
x<2
(-∞, 2)
Any number smaller than 2 is a solution for this inequality.
Example 2: Solve, graph, and write the interval notation
y+5>4
-5 -5
y
> -1
Subtract 5 from both sides
[-1, ∞)
-1 or any number larger than -1 is a solution for this inequality
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Remember, when multiplying or dividing by a negative number,
the value of the number will change from positive to negative or
negative to positive, i.e., the sign will change. When solving an
inequality, not only does the value of the number change, but the
inequality symbol will also change.
Example 3: Solve, graph, and write the interval notation
3(2x – 4) + 4x < 4(3x – 7) + 10
6x – 12 + 4x < 12x – 28 + 10 Distribute the 3 and the 4
10x – 12 < 12x – 18
Combine like terms
- 12x
- 12x
Subtract 12x from both sides
- 2x – 12 <
- 18
+ 12
+ 12
Add 12 to both sides
- 2x
<
-6
-2
-2
Divide both sides by -2
x
>
3
Notice < changed to >
(3, ∞)
Any number larger than 3 is a solution for this inequality.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)