Exercise 1 Page 458

a We are given a system of inequalities and are asked if the given ordered pairs are a solution.

y ≤ 3x+5 & (I) y ≥ - 23x+5 & (II) , Let's graph the system so that we can check the points.

Inequality (I)

Our first boundary line will be y=3x+5,

where 3 is the slope and 5 the y-intercept. Note that the boundary line will be solid because the inequality is less than or equal to.

To choose which side of the boundary line should be shaded, we will test a point. Let's use (0,0).

y≤3x+5

SubstituteII

x= 0, y= 0

0? ≤3( 0)+5

Multiply

0? ≤0+5

AddTerms

Add terms

0≤ 5

Because 0≤ 5, we will shade the region containing the point (0,0).

Inequality (II)

Our second boundary line will be y=- 2/3x+5, where the slope is - 23 and the y-intercept is 5. Note that this boundary line will also be solid because the inequality is greater than or equal to. To decide which region should be shaded, we will test again the point (0,0).

y≥- 2/3x+5

SubstituteII

x= 0, y= 0

0? ≥- 2/3( 0)+5

Multiply

0? ≥0+5

AddTerms

Add terms

0≱ 5

Because 0≱ 5, we will shade the region which does not contain the point (0,0).

Final solution

The possible solutions are where the shaded regions overlap.

Finally, we will plot the given points and see if they are located in the shaded area.

As we can see above, (- 1,4) is not inside the shaded area. This ordered pair is not a solution of the system.

b Looking at the final graph in Part A, we can see that this ordered pair, (0,5), is a solution of the system.

c Once again looking at the final graph from Part A, we can see that this ordered pair, (2,9), is a solution of the system.

Subchapter links

Exercises p.458

Exercise 1 Page 458

Hints

Check the answer

Inequality (I)

Inequality (II)

Final solution