Exercise 58 Page 152

Graphing an inequality involves two main steps.

1. Plotting the boundary line.
2. 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 5x+7y ≥ - 20 & 5x+7y = - 20To draw this line, we will first rewrite the equation in slope-intercept form.

5x+7y=- 20

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Write in slope-intercept form

SubEqn

LHS-5x=RHS-5x

7y=- 5x-20

DivEqn

.LHS /7.=.RHS /7.

y=-5/7x-20/7

Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept (0, b). y=-5/7x - 20/7 ⇔ y=-5/7x+( -20/7) We will plot the y-intercept ( 0, - 207), then use the slope m=- 57 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, 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 (0,0) as our test point.

5x+7y≥- 20

SubstituteII

x= 0, y= 0

5( 0)+7( 0)? ≥- 20

ZeroPropMult

Zero Property of Multiplication

0+0? ≥- 20

AddTerms

Add terms

0≥- 20

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