Begin by writing the inequalities. While you are writing the inequalities, you should remember that at least means that greater than or equal to and no more than means that less than or equal to.
Inequalities:1 ≤ x ≤ 4 y ≥ 10 15x+0.95y ≤ 63 Graph:
Practice makes perfect
Let x represent the number of audio books and y represent the number of songs that Steve puts on his smartphone. Steve wants at least 1 audio book but no more than 4. Thus, the number of audio books will be between 1 and 4.
1 ≤ x ≤ 4
Steve also wants at least 10 songs in his smartphone which means the number of songs will be greater than or equal to 10.
y ≥ 10
Next, we will write another inequality for the cost.
Verbal Expression
Algebraic Expression
Cost of x audio books ($)
15 x
Cost of y songs ($)
0.95 y
Total cost will be less than or equal to $ 63.
15 x+ 0.95 y ≤ 63
As a result, we have three inequalities to write a system.
1 ≤ x ≤ 4 & (I) y ≥ 10 & (II) 15x+0.95y ≤ 63 & (III)
We will graph the system starting from the first inequality. It is a compound inequality so it has two boundary lines. Let's indicate them!
1 ≤ x ≤ 4
Inequalities
Boundary Lines
x ≥ 1
x=1
x ≤ 4
x=4
The boundary lines are vertical lines that passes through the points (1,0) and (4,0). The inequality states that the points with x-coordinates between 1 and 4 are included in the solution set. Therefore, we will shade the region between the lines and lines will be solid because of the non-strict inequality.
The second inequality can be graphed in the same way.
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Inequality II &Boundary Line II
y ≥ 10 &y = 10
The boundary line is horizontal line and the points with y-coordinates greater than or equal to 10 are included in the solution set. Therefore, above the boundary line should be shaded.
The only inequality left is the third one.
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Inequality III &Boundary Line III
15x+0.95y ≤ 63 & 15x+0.95y = 63
The third boundary line is in standard form. Therefore, it would be a better way to find its intercepts to graph it. We will substitute y for 0 for the x-intercept and x for 0 for the y-intercept.
15x+0.95y=63
Operation
x-intercept
y-intercept
Substitution
15x+0.95( 0)=63
15( 0)+0.95y=63
Calculation
x=4.2
y=66.3
Point
(4.2,0)
(0,66.3)
Now we can plot the intercepts and connect them with a line segment.
Next, we will test the point (0,0) to decide the region that will be shaded.
