Linear Relations and Functions B-5 Graphing Inequalities

Objectives Graph linear inequalities Graph absolute value inequalities

Essential Question How do you determine which region to shade when graphing an inequality?

A linear inequality resembles a linear equation, but with an inequality symbol instead of an equality symbol. For example, y ≤ 2x + 1 is a linear inequality and y = 2x + 1 is the related linear equation.

The graph of y = 2x + 1 separates the coordinate plane into two regions. The line is the boundary of each region. The graph of the inequality y ≤ 2x + 1 is the shaded region. Every point in the shaded region satisfies the inequality. The graph of y = 2x + 1is drawn as a solid line to show that points on the line satisfy the inequality. If the inequality symbol were < or >, then points on the boundary would not satisfy the inequality, so the boundary would be drawn as a dashed line. y = 2x + 1 y ≤ 2x + 1

You can graph an inequality by following these steps: Determine whether the boundary should be solid or dashed. Graph the boundary. Choose a point not on the boundary and test it in the inequality. If a true inequality results, shade the region containing your test point. If a false inequality results, shade the other region.

Graph The boundary is the graph of Since the inequality symbol is <, the boundary will be dashed. The x-intercept is (4, 0) and the y-intercept is (0, −2). Example 7-1a

Shade the region that contains (0, 0). Graph Test (0, 0). Original inequality true Shade the region that contains (0, 0). Example 7-1b

Graph Answer: Example 7-1c

Inequalities can sometimes be used to model real-world situations. Education The SAT has two parts. One tutoring company advertises that it specializes in helping students who have a combined score on the SAT that is 900 or less. Write an inequality to describe the combined scores of students who are prospective tutoring clients. Let x be the first part of the SAT and let y be the second part. Since the scores must be 900 or less, use the  symbol. Example 7-2a

The 2nd part are less than 1st part and together or equal to 900. x y  900 Answer: Example 7-2b

Graph the inequality. Since the inequality symbol is , the graph of the related linear equation is solid. This is the boundary of the inequality. Example 7-2c

Graph the inequality. Test (0, 0). Original inequality true Example 7-2d

Graph the inequality. Shade the region that contains (0, 0). Since the variables cannot be negative, shade only the part in the first quadrant. Example 7-2e

Answer: Yes, this student fits the tutoring company’s guidelines. Does a student with a verbal score of 480 and a math score of 410 fit the tutoring company’s guidelines? The point (480, 410) is in the shaded region, so it satisfies the inequality. Answer: Yes, this student fits the tutoring company’s guidelines. Example 7-2f

Class Trip Two social studies classes are going on a field trip Class Trip Two social studies classes are going on a field trip. The teachers have asked for parent volunteers to also go on the trip as chaperones. However, there is only enough seating for 60 people on the bus. a. Write an inequality to describe the number of students and chaperones that can ride on the bus. Answer: Example 7-2g

c. Can 45 students and 10 chaperones go on the trip? Answer: b. Graph the inequality. c. Can 45 students and 10 chaperones go on the trip? Answer: Answer: yes Example 7-2h

You can define an absolute value function as 𝑓 𝑥 = 𝑥 and is the parent function for the family of all absolute value functions. The graph of 𝑓 𝑥 = 𝑥 is V-shaped and is symmetric about the y-axis. So, for every point (x, y) on the graph, the point (−x, y) is also on the graph. The highest or lowest point on the graph of an absolute value function is called the vertex. The vertex of the parent function is (0, 0).

You can derive new absolute value functions from the parent function through transformations of the parent graph. A transformation changes a graph’s size, shape, position, or orientation. A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape, or orientation.

The graph of 𝑦=𝑎 𝑥−ℎ +𝑘 is the graph of 𝑦= 𝑥 translated h units horizontally and k units vertically. The a represents a vertical stretch if 𝑎 >1 and is narrower. If 𝑎 <1, then the graph is compressed or wider than the parent graph. The vertex of 𝑦= 𝑥−ℎ +𝑘 is (h, k). When a = −1, the graph of 𝑦=𝑎 𝑥 is a reflection across the x-axis.

Graph Absolute Value Inequalities Graphing absolute value inequalities is similar to graphing linear inequalities. The inequality symbol determines whether the boundary is solid or dashed, and you can test a point to determine which region to shade.

Shade the region that contains (0, 0). Test (0, 0). Graph Since the inequality symbol is , the graph of the related equation is solid and has moved down 2 units from the parent graph. Graph the equation. Shade the region that contains (0, 0). Test (0, 0). Original inequality true Example 7-3a

Graph Answer: Example 7-3b

Essential Question How do you determine which region to shade when graphing an inequality? Choose a point not on the boundary and test it in the inequality. If a true inequality results, shade the region containing your test point. If a false inequality results, shade the other region. The point (0,0) is the easiest to use if it is not on the boundary.

Math Humor Teacher: Why are all your transformations in French? Student: They’re translations.