Did you know that in European hockey leagues, a player gets 2 points for a goal and 1 point for an assist? Suppose a player's contract stipulates that he receives a bonus if he gets more than 100 points. What linear inequality could you write to represent the number of goals and assists the players need to get to receive a bonus? How would you graph this inequality?

### Linear Inequalities with Two Variables

When a linear equation is graphed in a coordinate plane, the line splits the plane into two pieces. Each piece is called a **half plane**. The diagram below shows how the half planes are formed when graphing a linear equation.

A linear inequality in two variables can also be graphed. Instead of graphing only the **boundary line** **solution set** and is shown by shading, or coloring, the half plane that includes the appropriate solutions.

When graphing inequalities in two variables, you must remember when the value is included (

> The solution is the half plane above the line.≥ The solution is the half plane above the line and also all the points on the line.< The solution is the half plane below the line.≤ The solution is the half plane below the line and also all the points on the line.

The solution of

The solution of

The solution of

The solution of

This inequality is in slope-intercept form. Begin by graphing the line. Then determine the half plane to color.

- The inequality is
≥ , so the line is solid. - According to the inequality, you should shade the half plane above the boundary line.

In general, the process used to graph a linear inequality in two variables is:

**Step 1:** Graph the equation using the most appropriate method.

- Slope-intercept form uses the
y− intercept and slope to find the line. - Standard form uses the intercepts to graph the line.
- Point-slope uses a point and the slope to graph the line.

**Step 2:** If the equal sign is not included, draw a dashed line. Draw a solid line if the equal sign is included.

**Step 3:** Shade the half plane above the line if the inequality is “greater than.” Shade the half plane under the line if the inequality is “less than.”

#### Let's write an inequality and graph the solution for the following problem:

Julian has a job as an appliance salesman. He earns a commission of $60 for each washing machine he sells and $130 for each refrigerator he sells. How many washing machines and refrigerators must Julian sell to make $1,000 or more in commission?

Determine the appropriate variables for the unknown quantities. Let

Now translate the situation into an inequality.

Graph the standard form inequality using its intercepts. When

We want the ordered pairs that are solutions to Julian making more than $1,000, so we shade the half plane above the boundary line.

#### Graphing Horizontal and Vertical Linear Inequalities

Linear inequalities in one variable can also be graphed in the coordinate plane. They take the form of horizontal and vertical lines; however, the process is identical to graphing oblique, or slanted, lines.

#### Let's graph the inequality x>4 on 1) a number line and 2) the coordinate plane:

Remember what the solution to

The solution to this inequality is the set of all real numbers

On a coordinate plane, the line **does not equal** four, so the vertical line is dashed. This shows the reader that the ordered pairs on the vertical line

The inequality is looking for all

Graphing absolute value inequalities can also be done in the coordinate plane. To graph the inequality

Then graph each inequality on a coordinate plane.

In other words, the solution is all the coordinate points for which the value of

Both vertical lines are solid because points on the line are included in the solution.

### Examples

#### Example 1

Earlier, you were told that in European hockey leagues, a player gets 2 points for a goal and 1 point for an assist. Suppose a player's contract stipulates that he receives a bonus if he gets more than 100 points. What linear inequality could you write to represent the number of points and goals the players need to get a bonus? How would you graph this inequality?

Let's have

To graph this, you would graph the line

#### Example 2

A pound of coffee blend is made by mixing two types of coffee beans. One type costs $9.00 per pound and another type costs $7.00 per pound. Find all the possible mixtures of weights of the two different coffee beans for which the blend costs $8.50 per pound or less.

Begin by determining the appropriate letters to represent the varying quantities.

Let

Translate the information into an inequality:

Because the inequality is in standard form, it will be easier to graph using its intercepts.

When

Graph the inequality. The line will be solid. We shade below the line.

We graphed only the first quadrant of the coordinate plane because neither bag should have a negative weight.

The blue-shaded region tells you all the possibilities of the two bean mixtures that will give a total less than or equal to $8.50.

### Review

- Define half plane.
- In which cases would the boundary line be represented by a dashed line when graphing an inequality?
- In which cases would the boundary line be represented by a solid line when graphing an inequality?
- What is a method to help you determine which half plane to color when graphing an inequality?

In 5–26, graph each inequality in a coordinate plane.

x<20 y≥−5 y≤6 |x|>10 |y|≤7 y≤4x+3 y>−x2−6 y≤−12x+5 - \begin{align*}3x-4y \ge 12\end{align*}
3x−4y≥12 - \begin{align*}x+7y < 5\end{align*}
x+7y<5 - \begin{align*}y < -4x+4\end{align*}
y<−4x+4 - \begin{align*}y > \frac{7}{2}x+3\end{align*}
y>72x+3 - \begin{align*}6x+5y>1\end{align*}
6x+5y>1 - \begin{align*}6x-5y\le 15\end{align*}
6x−5y≤15 - \begin{align*}2x-y<5\end{align*}
2x−y<5 - \begin{align*}y+5 \le -4x+10\end{align*}
y+5≤−4x+10 - \begin{align*}x-\frac{1}{2}y \ge 5\end{align*}
x−12y≥5 - \begin{align*}y+4 \le -\frac{x}{3}+5\end{align*}
y+4≤−x3+5 - \begin{align*}5x-2y > 4\end{align*}
5x−2y>4 - \begin{align*}30x+5y < 100\end{align*}
30x+5y<100 - \begin{align*}y \ge -x\end{align*}
y≥−x - \begin{align*}6x-y < 4\end{align*}
6x−y<4 - Lili can make yarn ankle bracelets and wrist bracelets. She has 600 yards of yarn available. It takes 6 yards to make one wrist bracelet and 8 yards to make one ankle bracelet. Find all the possible combinations of ankle bracelets and wrist bracelets she can make without going over her available yarn.
- An ounce of gold costs $670 and an ounce of silver costs $13. Find all possible weights of silver and gold that make an alloy (combination of metals) that costs less than $600 per ounce.
- A phone company charges 50 cents per minute during the daytime and 10 cents per minute at night. How many daytime minutes and nighttime minutes would you have to use to pay more than $20.00 over a 24-hour period?
- Jesu has $30 to spend on food for a class barbeque. Hot dogs cost $0.75 each (including the bun) and burgers cost $1.25 (including bun and salad). Plot a graph that shows all the combinations of hot dogs and burgers he could buy for the barbecue, spending less than $30.00.
- At the local grocery store, strawberries cost $3.00 per pound and bananas cost $1.00 per pound. If I have $10 to spend between strawberries and bananas, draw a graph to show what combinations of each I can buy and spend at most $10.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.11.