a What is the total cost of decoration for a variable number of items? Can it exceed her budget?
B
b Identify the constraints and draw a graph for the inequality. Then check which part of the graph satisfies the inequality.
A
a 14x+60y ≤ 300
B
bExample Solutions: (8,2), (1,3), and (12,1)
Practice makes perfect
a She can spend at most $300 for the decoration of her bedroom. Let x represent the number of gallons of paint and y represent the number of sets of bed linens she purchases. We can write an expression to represent the total cost of these purchases.
Total Cost: 14x+60y
This cost should not exceed her budget. This constraint can be expressed as an inequality.
Inequality: 14x+60y ≤ 300
b In order to answer this part of the question we need to proceed in steps. We begin with graphing the boundary line and then we will test a point to shade the correct region.
Step 1: Boundary Line
To graph the boundary line, we treat the inequality as if it was an equation. Thus, we want to graph the following equation.
14x+60y = 300
Instead of rewriting it in a slope-intercept form, we can find its intercepts and connect the points with a straight line. We will substitute y= 0 for the x-intercept and x= 0 for the y-intercept.
14x+60y = 300
Operation
x-intercept
y-intercept
Substitution
14x+60( 0) = 300
14( 0)+60y = 300
Calculation
x≈ 21.4
y=5
Point
(21.4,0)
(0,5)
Let's plot the points and connect them with a line! Since the number of decoration items cannot be a negative number, the domain and range must contain non-negative numbers. Moreover, we will draw a solid boundary line, as the inequality is a non-strict inequality.
Step 2: Test Point
Let's choose (0,0) as the test point, as it is the most convenient one for the substitution.
We obtained a true statement. The point (0,0) is a part of the solution set.
Step 3: Graphing the Inequality
Since substituting (0,0) produced a true statement, we will shade the region that contains the point.
Step 4: Example Solutions
We can find an infinite number of ordered pairs that keep Sybrina within its budget because any point in the shaded region is a solution. The point (8,2), for example, means that she orders 8 gallons of paint and 2 sets of bed linens. Let's calculate total cost of these decorations.
8 * $14 + 2 * $60 & = $112 + $120
& = $232
This is less than $300 and keeps Sybrina within her budget.
Some other possible points are (1,3) and (12,1). Note that while Sybrina can purchase partial gallons of paint, it is not possible for her to buy a fraction of a bed linen set. Therefore, only points with whole number x-coordinates are acceptable.
