d All of the points that satisfy both inequalities.
Practice makes perfect
a The word at least is synonymous with minimum. Therefore, if a ruler sells for $1 and a compass sells for $2.50, we can write the following inequality. Remember that the club needs to make at least $15, so the sums must add up to at least that much.
1r+2.5c≥ 15 ⇔ r+2.5c≥ 15
b If we can sell a maximum of 25 items, the sum of rulers r and compasses c should be less than or equal to 25. With this information, we can write the following inequality.
r+c≤ 25
c To graph the inequalities with compasses on the x-axis and rulers on the y-axis, we first have to solve for r.
r+2.5c&≥ 15 ⇔ r≥ - 2.50c+ 15
r+c&≤ 25 ⇔ r≤ - c+ 25
We will begin by drawing the boundary lines. For this purpose, we will write them as equations.
r&= - 2.50c+ 15
r&= - c+ 25
Let's graph these lines. Note that the boundary lines are included, so we will make them solid lines.
To draw them as inequalities, we have to shade the correct side of the boundary line. By substituting a point that is not on any of the boundary lines into each inequality, we can check if the inequality should include the side that contains this point. Let's try the origin.
Point
Inequality
=
Shading
(0,0)
0? ≥ - 2.50( 0)+ 15
0 ≱ 15
Opposite side
(0,0)
0? ≤ - 0+ 25
0 ≤ 25
Same side
With this information we can complete the graph.
Notice that the region cannot fall below the x-axis or to the left of the y-axis, as this would mean we would sell either a negative number of rulers or compasses.
d The points in the solution region represent all of the points that satisfy both inequalities.
