Let x be the number of hours for writing a novel and y be the number of hours for exercising. Begin by writing the system of inequalities. Then determine the boundary line of each inequality to graph the system.
System:x+y≤ 35 7≤ y ≤ 15 20 ≤ x ≤ 25 Graph:
Practice makes perfect
Let x be the number of hours for writing a novel and y be the number of hours for exercising. Since Ramir has budgeted 35 per week, the sum of the number of hours will be less than or equal to 35.
x+ y ≤ 35
He wants to exercise at least 7 hours a week but no more than 15.
7≤ y ≤ 15
He is also planning to write between 20 and 25 hours per week.
20≤ x ≤ 25
Now we have three inequalities to write a system.
x+y≤ 35 & (I) 7≤ y ≤ 15 & (II) 20 ≤ x ≤ 25 & (III)
We will graph the system starting from the first inequality. First step to take is to determine the boundary line.
&Inequality I && Boundary Line I
&x+y ≤ 35 &&x+y = 35
The boundary line is in standard form. Therefore, it would be a better choice 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.
x+y=35
Operation
x-intercept
y-intercept
Substitution
x+ 0=35
0+y=35
Calculation
x=35
y=35
Point
(35,0)
(0,35)
Now we can plot the intercepts and connect them with a line segment. Notice that number of hours cannot be negative, so it will be bound by the axes. The boundary line will also be solid because of the non-strict inequality.
Next, we will decide which region we should shade. Therefore, we will test the point (0,0).
Since the point satisfies the inequality, region that contains the point will be shaded. Let's do it!
The first inequality has been graphed. Now we will graph the second inequality. Notice that the second inequality is a compound one. In this case there will be two boundary lines for it. Let's determine them!
7 ≤ y ≤ 15
Inequalities
Boundary Lines
y ≥ 7
y=7
y ≤ 15
y=15
The boundary lines of the second inequalities are horizontal lines that pass through the points (0,7) and (0,15). The inequality states that the points with y-values between 7 and 15 are included in the solution set. Therefore, the region between the lines will be shaded.
For the third inequality we will proceed in the same way.
20 ≤ x ≤ 25
Inequalities
Boundary Lines
x ≥ 20
x=20
x ≤ 25
x=25
The boundary lines are vertical lines that pass through the points (20,0) and (25,0). As in the second inequality, the region between the lines will be shaded.
The overlapping section of the graph above is the solution to the system.
