Graphing Linear Inequalities

**Graphing Linear Inequalities**

**linear inequality**occurs when the equal sign in a linear equation is replaced with an inequality symbol.

Examples of linear inequalities in two variables:

*y* __<__ 2*x* + 5

2*x* +5*y* < 7

4*x* + *y* __>__ -6

To graph a linear inequality in two variables, first graph the line as if it were a linear equation. Identify whether the inequality includes the "or equal to" aspect. Represent inequalities that do not include the possibility of "or equal to" with a dashed line and then shade the region that includes solutions to the inequality.

**Steps to Graphing Inequalities:**

Step 1 Change the inequality symbol to "=". Graph the equation. Use a dashed line for < or >. Use a solid line for __<__ or __>__.

Step 2 Test a point that is not on the line to check whether it is a solution of the inequality.

Step 3 If the test point is a solution, shade its region. If the test point is not a solution, shade the other region.

**Example**

Graph: *y* - 2*x* > 3

First change > to = and write the equation in slope-intercept form:

*y* - 2*x* = 3

*y* = 2*x* + 3

The slope is 2 and the *y*-intercept is 3. Graph the line that passes through point (0, 3) and has a slope of 2. The line needs to be dashed because this is an inequality.

Using (0, 0) as a test point we need to find out which side provides a solution:

*y* > 2*x* + 3

0 >2(0) + 3

0 3

The inequality doesn't hold, so (0, 0) is not a solution. We then shade the half-plane that does not contain (0, 0).

You can apply the information from a graph of an inequality to extrapolate data (that is, get information for data points that aren't explicitly mentioned) or to answer questions.

**Example**

You are buying art supplies for your art club. You have $40 to spend. Tubes of paint cost $6 each, and brushes cost $4 each. This situation can be expressed with the inequality 6*x* + 4*y* __<__ 40, where *x* represents the number of tubes of paint, and *y* represents the number of brushes.

How many tubes of paint and how many brushes can you buy with $40?

**Solution**

Graph the inequality where the equation of the line is *y* = -^{3}/_{2}*x* + 10, and test it at point (0, 0), giving us

6(0) + 4(0) __<__ 40

0 __<__ 40

The inequality holds, so (0, 0) is a solution. We then shade the half-plane that contains (0, 0).

b. Use the graph to find a solution, then interpret the solution.

One solution is (4, 4).

This means that you can buy 4 tubes of paint and 4 brushes.