Question

posttest: rational relationships
15
select the correct answer.
it takes greg and nikki a total of 40 minutes to paint a room if they work together. if greg works alone, he takes 18 minutes longer to paint the room than nikki takes if she works alone.
when x represents the number of minutes it takes nikki to paint the room working alone, the situation is represented by this equation:
\\(\frac{1}{x} + \frac{1}{x + 18} = \frac{1}{40}\\)
which value is a solution of the rational equation that must be discarded because of the context?
a. \\(x = 0\\)
b. \\(x = -10\\)
c. \\(x = -18\\)
d. \\(x = -72\\)

Explanation:

Step1: Analyze the context

In the problem, \( x \) represents the number of minutes it takes Nikki to paint the room alone. Time cannot be negative or zero (since zero minutes would mean she paints the room instantaneously, which is not realistic, and negative time has no meaning in this context).

Step2: Evaluate each option

- Option A: \( x = 0 \) is invalid because time cannot be zero (she can't paint a room in 0 minutes). But let's check other options too.
- Option B: \( x=- 10 \), negative time is invalid, but let's see the equation's domain. The original equation \(\frac{1}{x}+\frac{1}{x + 18}=\frac{1}{40}\) has restrictions \(x eq0\) and \(x eq - 18\) (to avoid division by zero). But in the context of time, \(x\) (Nikki's time) and \(x + 18\) (Greg's time) must be positive. So \(x>0\) and \(x+18>0\) (which is always true if \(x>0\)). Now, let's solve the equation \(\frac{1}{x}+\frac{1}{x + 18}=\frac{1}{40}\). Multiply through by \(40x(x + 18)\) to get \(40(x + 18)+40x=x(x + 18)\). Expand: \(40x+720 + 40x=x^{2}+18x\). Combine like terms: \(x^{2}+18x-80x - 720=0\), \(x^{2}-62x - 720=0\). Factor: \((x - 72)(x+10)=0\). Solutions are \(x = 72\) and \(x=-10\). But \(x\) represents time, so \(x=-10\) is negative and must be discarded. Wait, but let's check the options again. Wait, the options are \(x = 0\), \(x=-10\), \(x=-18\), \(x=-72\). Wait, when we solve the equation, we get \(x = 72\) and \(x=-10\). But let's check the context again. The time \(x\) (Nikki's time) must be positive. Also, \(x+18\) (Greg's time) must be positive. So \(x>0\). Now, let's check each option for context (time can't be negative or zero):
- \(x = 0\): invalid (time can't be zero)
- \(x=-10\): invalid (negative time)
- \(x=-18\): if \(x=-18\), then \(x + 18=0\), which is a division by zero in the original equation, but also negative time.
- \(x=-72\): negative time.

But the equation's solutions from solving are \(x = 72\) and \(x=-10\). But the question is which value is a solution of the rational equation (so it satisfies the equation) that must be discarded because of the context. Let's check if \(x=-10\) is a solution. Plug \(x=-10\) into the left - hand side: \(\frac{1}{-10}+\frac{1}{-10 + 18}=\frac{-1}{10}+\frac{1}{8}=\frac{-4 + 5}{40}=\frac{1}{40}\), which is equal to the right - hand side. So \(x=-10\) is a solution of the equation but must be discarded because time can't be negative. Wait, but let's check \(x=-72\): plug \(x=-72\) into the left - hand side: \(\frac{1}{-72}+\frac{1}{-72 + 18}=\frac{-1}{72}+\frac{1}{-54}=\frac{-3-4}{216}=\frac{-7}{216} eq\frac{1}{40}\). So \(x=-72\) is not a solution. \(x=-18\): plug \(x=-18\), we get division by zero in \(\frac{1}{x}\) and \(\frac{1}{x + 18}\) (since \(x+18 = 0\)), so it's not a solution. \(x = 0\): division by zero in \(\frac{1}{x}\), so not a solution. So the only solution of the equation (other than \(x = 72\)) is \(x=-10\), which must be discarded because time can't be negative. Wait, but let's re - solve the equation properly.

Starting with \(\frac{1}{x}+\frac{1}{x + 18}=\frac{1}{40}\)

Multiply both sides by \(40x(x + 18)\):

\(40(x + 18)+40x=x(x + 18)\)

\(40x+720+40x=x^{2}+18x\)

\(80x + 720=x^{2}+18x\)

\(x^{2}+18x-80x - 720=0\)

\(x^{2}-62x - 720=0\)

Using the quadratic formula \(x=\frac{62\pm\sqrt{62^{2}-4\times1\times(-720)}}{2}=\frac{62\pm\sqrt{3844 + 2880}}{2}=\frac{62\pm\sqrt{6724}}{2}=\frac{62\pm82}{2}\)

We get \(x=\frac{62 + 82}{2}=\frac{144}{2}=72\) and \(x=\frac{62-82}{2}=\frac{-20}{2}=-10\)

So the solutions of the equation are \(x = 72\) and \(x=-10\). Since \(x\) represents time, \(x=-10\) is a solution of the equation but must be discarded because time cannot be negative. \(x = 0\) is not a solution (division by zero), \(x=-18\) is not a solution (division by zero in \(x + 18\)), \(x=-72\) is not a solution (as we saw earlier).

Answer:

B. \(x=-10\)