Use each item to define a variable. In terms of the variables, how much will you pay for your purchase? Remember that this amount must be less than $45.
Inequality: x+5y < 45 Graph:
Practice makes perfect
We need to translate the given information into mathematical expressions. First, notice that the sentence All the items cost $1 or $5 tells us that there are two types of items at the garage sale. Therefore, we need to define two variables.
x = number of $1 items bought
y = number of $5 items bought
The total amount of money we need to pay for our purchases at the garage sale is shown in the following table.
Verbal Expression
Algebraic Expression
Price of $1 items
1(x)=x
Price of $5 items
5(y)=5y
Total ($)
x+5y
Now, the phrase you spend less than $45 implies that the total amount must be less than $45, which leads us to the following inequality.
x+5y < 45
In order to graph our inequality on a coordinate plane, let's start by finding the boundary line. This can be done by replacing the inequality symbol with an equals sign.
Inequality &&&& Boundary Line
x+5y < 45 &&&& x+5y = 45
Since this line is not in slope-intercept form, let's rewrite it by isolating the variable y.
The boundary line has a slope of - 15 and the y-intercept is 9. Moreover, since the inequality is strict, the line will be dashed.
To know which region we have to shade, let's test (0,0) in the inequality. If we get a true statement, we will shade the region that contains the origin. Otherwise, we will shade the region that does not contain it.
Since 0 is less than 45, we got a true statement. Hence, we will shade the region that contains the origin.
Watch out! The entirety of the region above does not represent the given situation. Remember that x and y represent the number of items bought, so they cannot be negative. Therefore, the region needs to be limited to the first quadrant, including the axes.
Alternative Solution
Alternative Inequality
Notice that if you choose x to be the number of $5 items bought and y to be the number of $1 items bought, then the expression that represents the amount of money we have to pay will be slightly different.
5x+y
Then, the inequality will be 5x+y<45. Well, this is nothing to worry about. Both inequalities are totally equivalent. The solutions will represent the same information.
