Exercise 45 Page 509

Practice makes perfect

a We want to match each graph with the correct inequality. To do it we will consider each inequality one at a time, graph it, and then match with one of the given graphs. We will begin with with the first inequality y> - x+2. Graphing an inequality involves two main steps.

Plotting the boundary line.
Shading half of the plane to show the solution set.

Boundary Line

To graph the inequality we have to draw the boundary line. The equation of a boundary line is written by replacing the inequality symbol from the inequality with an equals sign.

Inequality & Boundary Line y > - x+2 & y = - x+2 Fortunately this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b). y=-1x+ 2 We will plot the y-intercept (0, 2), then use the slope m=-1 to plot another point on the line. Connecting these points with a dashed line will give us the boundary line of our inequality. Note that the boundary line is dashed, not solid, because the inequality is strict.

Shading the Plane

To decide which side of the boundary line to shade, we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use (0,0) as our test point.

y>- x+2

SubstituteII

x= 0, y= 0

0 ? > - 0 + 2

AddTerms

Add terms

0 ≯ 2

Since the substitution of the test point did not create a true statement, we will shade the region that does not contain the point.

Notice that the graph we obtained corresponds to the given Graph 3.

b Now, we will graph the second inequality y<2x-3 and match it with one of the given graphs.

Boundary Line

To graph the inequality we have to draw the boundary line. The equation of a boundary line is written by replacing the inequality symbol from the inequality with an equals sign. Inequality & Boundary Line y < 2x-3 & y = 2x-3

Fortunately this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b). y=2x-3 ⇔ y=2x+( - 3) We will plot the y-intercept (0, - 3), then use the slope m=2 to plot another point on the line. Connecting these points with a dashed line will give us the boundary line of our inequality. Note that the boundary line is dashed, not solid, because the inequality is strict.

Shading the Plane

y<2x-3

SubstituteII

x= 0, y= 0

0 ? < 2( 0)-3

▼

Evaluate

ZeroPropMult

Zero Property of Multiplication

0 ? < 0-3

SubTerm

Subtract term

0 ≮ - 3

Since the substitution of the test point did not create a true statement, we will shade the region that does not contain the point.

Notice that the graph we obtained corresponds to Graph 1.

c Now, we will graph y ≥ 12x.

Boundary Line

Fortunately this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b). y=1/2x+ 0 We will plot the y-intercept (0, 0), then use the slope m=12 to plot another point on the line. Connecting these points with a solid line will give us the boundary line of our inequality. Note that the boundary line is solid, not dashed, because the inequality is not strict.

Shading the Plane

To decide which side of the boundary line to shade we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use (1,1) as our test point.

y ≥ 1/2

SubstituteII

x= 1, y= 1

1 ? ≥ 1/2( 1)

IdPropMult

Identity Property of Multiplication

1 ≥ 1/2

Since the substitution of the test point created a true statement, we will shade the region that contains the point.

Notice that the graph we obtained corresponds to Graph 4.

d Finally, we will graph the last inequality y ≤ - 23x+2.

Boundary Line

Fortunately this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b). y=- 2/3x+ 2 We will plot the y-intercept (0, 2), then use the slope m=- 23 to plot another point on the line. Connecting these points with a solid line will give us the boundary line of our inequality. Note that the boundary line is solid, not dashed, because the inequality is not strict.