Begin by writing a system of inequalities for the situation. You should remember that no more than means less than or equal to and at least means greater than or equal to.
Practice makes perfect
We will begin by writing a system of inequalities. Let x be the number of hours and y be the total cost of recording. We should remember that no more than means less than or equal to and at least means greater than or equal to.
Verbal Expression
Algebraic Expression
Cost of x hours of recording ($)
35 x
Total cost is less than or equal to $575.
y ≤ 575
Total cost is greater than or equal to $35 x.
y ≥ 35 x
Now we have to inequalities to write a system.
y ≤ 575 & (I) y ≥ 35x & (II)
Let's begin by graphing Inequality I. We will first find the boundary line by replacing the inequality sign with the equals sign.
&Inequality I &&Boundary Line I
&y ≤ 575 && y = 575The boundary line of the inequality is a horizontal line that passes through the point (0,575). Since the number of hours and the cost cannot be negative the line will be bound by the axes. It will also be solid because the inequality is non-strict. Let's draw it!
The inequality says that the points with the y-coordinates less than or equal to 575 are included in the solution set. Therefore, we will shade region below the boundary line.
We will graph the second inequality proceeding in the same way.
&Inequality II &&Boundary Line II
&y ≥ 35x && y = 35x
The boundary line is in the slope-intercept form so that we can immediately identify its slope and y-intercept.
y=35x+
We will first plot the y-intercept and find a second point using the slope. Thus, we will be able to draw a line.
Now we can draw the line. Since the inequality is non-strict, the boundary line will be solid.
To complete the inequality, we should decide which region we should shade. To do so, we will test an arbitrary point that is not on the boundary line. Let the point (4,400) be our test point.
