Linear Inequalities in Two Variables

Graph inequalities like 2x + 3 < y on the coordinate plane

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Linear Inequalities in Two Variables

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 \begin{align*}(y=mx+b)\end{align*}, you must also include all the other ordered pairs that could be solutions to the inequality. This is called the 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 (\begin{align*}\le\end{align*} or \begin{align*}\ge\end{align*}) or not included (\begin{align*} < \end{align*} or \begin{align*} > \end{align*}). To represent these inequalities on a coordinate plane, instead of shaded or unshaded circles, we use solid and dashed lines. We can tell which half of the plane the solution is by looking at the inequality sign.

• \begin{align*} > \end{align*} The solution is the half plane above the line.
• \begin{align*}\ge\end{align*} The solution is the half plane above the line and also all the points on the line.
• \begin{align*} < \end{align*} The solution is the half plane below the line.
• \begin{align*}\le\end{align*} The solution is the half plane below the line and also all the points on the line.

The solution of \begin{align*}y > mx+b\end{align*} is the half plane above the line. The dashed line shows that the points on the line are not part of the solution.

The solution of \begin{align*}y \ge mx+b\end{align*} is the half plane above the line and all the points on the line.

The solution of \begin{align*}y < mx+b\end{align*} is the half plane below the line.

The solution of \begin{align*}y \le mx+b\end{align*} is the half plane below the line and all the points on the line.

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

• The inequality is \begin{align*}\ge\end{align*}, 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 \begin{align*}y-\end{align*}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 \begin{align*}x =\end{align*} number of washing machines Julian sells and let \begin{align*}y =\end{align*} number of refrigerators Julian sells. Now translate the situation into an inequality. \begin{align*}60x + 130y \ge 1,000\end{align*}. Graph the standard form inequality using its intercepts. When \begin{align*}x=0, y=7.692\end{align*}. When \begin{align*}y=0, x=16.667\end{align*}. The line will be solid. We want the ordered pairs that are solutions to Julian making more than1,000, 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 \begin{align*}x>4\end{align*} on 1) a number line and 2) the coordinate plane:

Remember what the solution to \begin{align*}x>4\end{align*} looks like on a number line.

The solution to this inequality is the set of all real numbers \begin{align*}x\end{align*} that are larger than four but not including four.

On a coordinate plane, the line \begin{align*}x=4\end{align*} is a vertical line four units to the right of the origin. The inequality does not equal four, so the vertical line is dashed. This shows the reader that the ordered pairs on the vertical line \begin{align*}x=4\end{align*} are not solutions to the inequality.

The inequality is looking for all \begin{align*}x-\end{align*}coordinates larger than four. We then color the half plane to the right, symbolizing \begin{align*}x>4\end{align*}.

Graphing absolute value inequalities can also be done in the coordinate plane. To graph the inequality \begin{align*}|x|\ge 2\end{align*}, we can recall a previous Concept and rewrite the absolute value inequality.

\begin{align*}x \le -2\end{align*} or \begin{align*}x \ge 2\end{align*}

Then graph each inequality on a coordinate plane.

In other words, the solution is all the coordinate points for which the value of \begin{align*}x\end{align*} is smaller than or equal to –2 and greater than or equal to 2. The solution is represented by the plane to the left of the vertical line \begin{align*}x=-2\end{align*} and the plane to the right of line \begin{align*}x=2\end{align*}.

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 \begin{align*}x\end{align*} represent the number of goals and \begin{align*}y\end{align*} represent the number of assists. Then, the inequality that represents this situation is:

\begin{align*}2x + y \ge 100\end{align*}

To graph this, you would graph the line \begin{align*}2x + y = 100\end{align*} using the intercepts of (50, 0) and (0, 100). The line would be solid and the shaded area would be above the line because it is greater than or equal.

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 \begin{align*}x =\end{align*} weight of9.00 per pound coffee beans in pounds and let \begin{align*}y =\end{align*} weight of 7.00 per pound coffee beans in pounds. Translate the information into an inequality: \begin{align*}9x+7y \le 8.50\end{align*} Because the inequality is in standard form, it will be easier to graph using its intercepts. When \begin{align*}x=0, y=1.21\end{align*}. When \begin{align*}y=0, x=0.944\end{align*}. 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 to8.50.

Review

1. Define half plane.
2. In which cases would the boundary line be represented by a dashed line when graphing an inequality?
3. In which cases would the boundary line be represented by a solid line when graphing an inequality?
4. 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.

1. \begin{align*}x < 20\end{align*}
2. \begin{align*}y \ge -5\end{align*}
3. \begin{align*}y \le 6\end{align*}
4. \begin{align*}|x| > 10\end{align*}
5. \begin{align*}|y| \le 7 \end{align*}
6. \begin{align*}y \le 4x+3\end{align*}
7. \begin{align*}y > -\frac{x}{2}-6\end{align*}
8. \begin{align*}y\le -\frac{1}{2} x+5\end{align*}
9. \begin{align*}3x-4y \ge 12\end{align*}
10. \begin{align*}x+7y < 5\end{align*}
11. \begin{align*}y < -4x+4\end{align*}
12. \begin{align*}y > \frac{7}{2}x+3\end{align*}
13. \begin{align*}6x+5y>1\end{align*}
14. \begin{align*}6x-5y\le 15\end{align*}
15. \begin{align*}2x-y<5\end{align*}
16. \begin{align*}y+5 \le -4x+10\end{align*}
17. \begin{align*}x-\frac{1}{2}y \ge 5\end{align*}
18. \begin{align*}y+4 \le -\frac{x}{3}+5\end{align*}
19. \begin{align*}5x-2y > 4\end{align*}
20. \begin{align*}30x+5y < 100\end{align*}
21. \begin{align*}y \ge -x\end{align*}
22. \begin{align*}6x-y < 4\end{align*}
23. 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.
24. 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. 25. 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?
26. 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.
27. 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.

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

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Color Highlighted Text Notes

Vocabulary Language: English Spanish

TermDefinition
boundary line The boundary line is an equation $(y=mx+b)$ that creates two half planes.
half plane 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.
solution set All ordered pairs that could be solutions to an inequality, which are shown graphically by shading, or coloring, the half plane that includes the appropriate solutions.
Cartesian Plane The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.
Linear Inequality Linear inequalities are inequalities that can be written in one of the following four forms: $ax + b > c, ax + b < c, ax + b \ge c$, or $ax + b \le c$.
Slope-Intercept Form The slope-intercept form of a line is $y = mx + b,$ where $m$ is the slope and $b$ is the $y-$intercept.

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