Question

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 $20x + 25y \geq 200$ and $x + y < 10$ represent the situation. which is true of the graph of the solution set? check all that apply.\
\
\\(\square\\) the line $20x + 25y \geq 200$ has a positive slope and a negative $y$-intercept.\
\\(\square\\) the line $x + y < 10$ has a negative slope and a positive $y$-intercept.\
\\(\square\\) the line representing $20x + 25y \geq 200$ is solid and the graph is shaded above the line.\
\\(\square\\) the line representing $x + y < 10$ is dashed and the graph is shaded above the line.\
\\(\square\\) the overlapping region contains the point $(4, 5)$.\

Explanation:

Step1: Analyze \(20x + 25y \geq 200\)

Rewrite in slope - intercept form \(y=mx + b\) (\(m\) is slope, \(b\) is \(y\) - intercept):
\(25y\geq - 20x + 200\)
\(y\geq-\frac{20}{25}x + 8\)
\(y\geq-\frac{4}{5}x + 8\)
Slope \(m =-\frac{4}{5}\) (negative), \(y\) - intercept \(b = 8\) (positive). The inequality is \(\geq\), so the line is solid and shaded above the line (since \(y\) is greater than or equal to the line).

Step2: Analyze \(x + y\lt10\)

Rewrite in slope - intercept form: \(y\lt - x+10\)
Slope \(m=- 1\) (negative), \(y\) - intercept \(b = 10\) (positive). The inequality is \(\lt\), so the line is dashed and shaded below the line (since \(y\) is less than the line).

Step3: Check each option

• Option 1: The line \(20x + 25y\geq200\) has slope \(-\frac{4}{5}\) (negative) and \(y\) - intercept \(8\) (positive). So this option is false.
• Option 2: The line \(x + y\lt10\) (or \(y\lt - x + 10\)) has slope \(- 1\) (negative) and \(y\) - intercept \(10\) (positive). So this option is true.
• Option 3: The line for \(20x + 25y\geq200\) is solid (because of \(\geq\)) and shaded above the line (because \(y\geq\) the line). So this option is true.
• Option 4: The line for \(x + y\lt10\) is dashed (because of \(\lt\)) but shaded below the line (because \(y\lt\) the line), not above. So this option is false.
• Option 5: Check if \((4,5)\) is in the overlapping region.

For \(20x + 25y\geq200\): \(20\times4+25\times5=80 + 125=205\geq200\) (satisfies).
For \(x + y\lt10\): \(4 + 5=9\lt10\) (satisfies). So \((4,5)\) is in the overlapping region. This option is true.

Answer:

B. The line \(x + y\lt10\) has a negative slope and a positive \(y\) - intercept.
C. The line representing \(20x + 25y\geq200\) is solid and the graph is shaded above the line.
E. The overlapping region contains the point \((4,5)\)