Using inequalities to determine the appearance of a graph and solution set:
As a computer technician, Andre makes $20 per hour to diagnose a problem and $25 per hour to fix a problem. He works fewer than 10 hours per week but wants to make at least $200 per week. The inequalities [tex]$20 x+25 y \geq 200$[/tex] and [tex]$x+y < 10$[/tex] represent the situation. Which is true of the graph of the solution set? Check all that apply.
A. The line [tex]$20 x+25 y \geq 200$[/tex] has a positive slope and a negative y-intercept.
B. The line [tex]$x+y < 10$[/tex] has a negative slope and a positive y-intercept.
C. The line representing [tex]$20 x+25 y \geq 200$[/tex] is solid, and the graph is shaded above the line.
D. The line representing [tex]$x+y < 10$[/tex] is dashed, and the graph is shaded above the line.
E. The overlapping region contains the point (4,5).
Answer (2)
Rewrite the inequalities in slope-intercept form to find the slope and y-intercept of each line.
Determine if each line is solid or dashed based on the inequality sign, and whether the graph is shaded above or below the line.
Check if the point ( 4 , 5 ) satisfies both inequalities to determine if it lies in the overlapping region.
The correct statements are: The line x + y < 10 has a negative slope and a positive y -intercept; The line representing 20 x + 25 y ≥ 200 is solid and the graph is shaded above the line; The overlapping region contains the point ( 4 , 5 ) .
Explanation
Problem Analysis We are given two inequalities: 20 x + 25 y ≥ 200 and x + y < 10 . We need to analyze the properties of their graphs and determine which of the given statements are true.
Analyzing Inequality 1 Let's analyze the first inequality, 20 x + 25 y ≥ 200 . We can rewrite this in slope-intercept form ( y = m x + b ) to easily identify the slope and y-intercept. Subtracting 20 x from both sides gives 25 y ≥ − 20 x + 200 . Dividing by 25 gives y ≥ − 5 4 x + 8 . The slope is − 5 4 , which is negative, and the y-intercept is 8, which is positive. Since the inequality is ≥ , the line is solid, and the region is shaded above the line.
Analyzing Inequality 2 Now let's analyze the second inequality, x + y < 10 . Rewriting this in slope-intercept form, we get y < − x + 10 . The slope is − 1 , which is negative, and the y-intercept is 10, which is positive. Since the inequality is < , the line is dashed, and the region is shaded below the line.
Checking the Statements Now, let's check the given statements:
"The line 20 x + 25 y ≥ 200 has a positive slope and a negative y -intercept." This is false. The slope is − 5 4 (negative) and the y-intercept is 8 (positive).
"The line x + y < 10 has a negative slope and a positive y -intercept." This is true. The slope is − 1 (negative) and the y-intercept is 10 (positive).
"The line representing 20 x + 25 y ≥ 200 is solid and the graph is shaded above the line." This is true because the inequality is ≥ .
"The line representing x + y < 10 is dashed and the graph is shaded above the line." This is false. The line is dashed, but the graph is shaded below the line because the inequality is < .
"The overlapping region contains the point ( 4 , 5 ) ." Let's check if the point ( 4 , 5 ) satisfies both inequalities:
20 ( 4 ) + 25 ( 5 ) = 80 + 125 = 205 ≥ 200 . This is true.
4 + 5 = 9 < 10 . This is true. Since both inequalities are satisfied, the point ( 4 , 5 ) is in the overlapping region.
Examples
Understanding inequalities and their graphical representation is crucial in various real-world applications. For instance, in business, companies use inequalities to model constraints on resources and production. Imagine a bakery that makes cakes and cookies. They have limited amounts of flour and sugar. Inequalities can help them determine the possible combinations of cakes and cookies they can produce while staying within their resource constraints. The solution set of these inequalities, when graphed, shows all feasible production plans. This helps the bakery optimize their production to maximize profit.
The true statements regarding the inequalities representing Andre's work situation are B (the line x + y < 10 has a negative slope and a positive y-intercept), C (the line 20 x + 25 y ≥ 200 is solid and shaded above), and E (the overlapping region contains the point (4, 5)).
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