Ryan Zimmerman
08/17/2024 · Primary School
Consider \( g(x)=5 x-12 \). For what interval(s) is \( g(x) \) positive? negative? positive: \( \left(\frac{12}{5}, \infty\right) \); negative: \( \left(-\infty, \frac{12}{5}\right) \) o positive: \( (-\infty,-12) \); negative: \( (-12, \infty) \) positive: \( (-12, \infty) \); negative: \( (\infty,-12) \) positive: \( \left(-\infty, \frac{5}{12}\right) \); negative: \( \left(\frac{5}{12}, \infty\right) \)
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Tutor-Verified Answer
Step-by-step Solution
To determine the intervals where the function \( g(x) = 5x - 12 \) is positive or negative, we first need to find the value of \( x \) where \( g(x) = 0 \).
Setting \( g(x) \) to zero:
\[
5x - 12 = 0
\]
Solving for \( x \):
\[
5x = 12 \\
x = \frac{12}{5}
\]
Now, we can analyze the sign of \( g(x) \) in the intervals determined by this critical point \( x = \frac{12}{5} \).
1. **For \( x < \frac{12}{5} \)**:
- Choose a test point, for example, \( x = 0 \):
\[
g(0) = 5(0) - 12 = -12 \quad (\text{negative})
\]
2. **For \( x > \frac{12}{5} \)**:
- Choose a test point, for example, \( x = 3 \):
\[
g(3) = 5(3) - 12 = 15 - 12 = 3 \quad (\text{positive})
\]
From this analysis, we conclude:
- \( g(x) \) is **negative** for \( x < \frac{12}{5} \) (i.e., in the interval \( (-\infty, \frac{12}{5}) \)).
- \( g(x) \) is **positive** for \( x > \frac{12}{5} \) (i.e., in the interval \( \left(\frac{12}{5}, \infty\right) \)).
Thus, the correct intervals are:
- Positive: \( \left(\frac{12}{5}, \infty\right) \)
- Negative: \( \left(-\infty, \frac{12}{5}\right) \)
So the answer is:
**Positive: \( \left(\frac{12}{5}, \infty\right) \); Negative: \( \left(-\infty, \frac{12}{5}\right) \)**.
Quick Answer
Positive: \( \left(\frac{12}{5}, \infty\right) \); Negative: \( \left(-\infty, \frac{12}{5}\right) \)
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