Question

The graphs of f(x) and g(x) are shown below.

On a coordinate plane, a straight line with a positive slope represents f (x) = x minus 3. The line goes through (0, negative 3) and (3, 0). On a coordinate plane, a straight line with a negative slope represents g (x) = negative 0.5 x. The line goes through (negative 4, 2), (0, 0), and (4, negative 2)

For what interval is the value of (f – g)(x) negative?
(negative infinity, negative 1)
(negative infinity, 2)
(0, 3)
(2, infinity)

Answer 1

To find the interval where \((f - g)(x) < 0\), we first need to express \((f - g)(x)\) explicitly.

Given:

- \(f(x) = x - 3\)
- \(g(x) = -0.5x\)

Now, the expression for \((f - g)(x)\) is:

\[
(f - g)(x) = f(x) - g(x) = (x - 3) - (-0.5x)
\]

This simplifies to:

\[
(f - g)(x) = x - 3 + 0.5x = 1.5x - 3
\]

Next, we need to find when this expression is negative:

\[
1.5x - 3 < 0
\]

Solving for \(x\):

\[
1.5x < 3
\]

\[
x < \frac{3}{1.5} = 2
\]

Thus, \((f - g)(x) < 0\) for \(x < 2\).

Now we consider the intervals given:

- \((-\infty, -1)\)
- \((-\infty, 2)\)
- \((0, 3)\)
- \((2, \infty)\)

The interval where \((f - g)(x)\) is negative is:

\[
\text{The correct answer is } (-\infty, 2).
\]