Exercise 45 Page 254

To begin, let's list the given information.

• The points (- 7, - 16) and (1,8) lie on the boundary line. The boundary line must pass through these points.
• The points (- 7, 0) and (3,14) are not solutions of the inequality. These points cannot lie in the shaded region on the graph.

Let's plot the given points on a coordinate plane.

To write the inequality we must complete the following.

1. Draw the boundary line.
2. Shade the region containing the solutions.

   Drawing the Boundary Line

   As was established above, we must draw the boundary line so that points (-7,-16) and (1,8) lie on the line. First, we will determine the slope of the line using the Slope Formula.

   m = y_2-y_1/x_2-x_1

   SubstitutePoints

   Substitute ( - 7,-16) & ( 1,8)

   m= 8-( - 16)/1-( - 7)

   ▼

   Simplify

   NegNeg

   - (- a)=a

   m= 8+16/1+7

   AddTerms

   Add terms

   m=24/8

   CalcQuot

   Calculate quotient

   m=3

   If we write our equation in the form y=mx+b, we now have the partial equation. y=3x+b To determine the value of b we can substitute the coordinates of (-7,-16) or (1,8) into the above equation for x and y and solve. We will arbitrarily choose to use the coordinates of (1,8).

   y=3x+b

   ▼

   Solve for b

   SubstituteII

   x= 1, y= 8

   8=3 * 1 +b

   Multiply

   Multiply

   8=3+b

   SubEqn

   LHS-3=RHS-3

   5=b

   RearrangeEqn

   Rearrange equation

   b=5

   Now we can complete the equation for the boundary line. y=3x+5 Let's add the boundary line to the graph above.

   

   We can see that (-7,-16) and (1,8) lie on the boundary line. However, (3,14) also lies on the line. Since (3,14) is not a solution of the inequality, we must dash the boundary line. This means the inequality symbol will either be < or >. We will determine that next.

   

   Now, we can determine which region of the coordinate plane to shade.

   Shading the Inequality

   At this point, we have the following inequality. y 3x+5 From the graph above, we can see that point (-7,0) lies to the left of the boundary line. Since(-7,0) is not a solution, we want to shade the region to the right. To determine whether < or > is the correct inequality symbol, we can choose any point from the region on the right and substitute it into the above inequality. We will arbitrarily choose the point (0,0).

   y ? 3x+5

   SubstituteII

   x= 0, y= 0

   0 ? 3 * 0 +5

   ZeroPropMult

   Zero Property of Multiplication

   0 < 5

   To ensure that (0,0) is a solution of the inequality, the symbol must be <. Now we can complete our inequality. y<3x+5 We will graph the inequality to ensure that we have satisfied the requirements.