Sure! Let's carefully analyze and solve the given inequality [tex]\( f(x) < g(x) \)[/tex] with the constraints provided.
### Given Functions
1. [tex]\( f(x) = \frac{4}{x-3} + 2 \)[/tex]
2. [tex]\( g(x) = x + 2 \)[/tex]
### Inequality to Solve
We need to solve:
[tex]\[ \frac{4}{x-3} + 2 < x + 2 \][/tex]
### Step-by-Step Solution
1. Subtract 2 from both sides:
[tex]\[
\frac{4}{x-3} + 2 - 2 < x + 2 - 2
\][/tex]
This simplifies to:
[tex]\[
\frac{4}{x-3} < x
\][/tex]
2. Isolate the fraction:
To isolate the fraction [tex]\(\frac{4}{x-3}\)[/tex] on one side of the inequality:
[tex]\[
\frac{4}{x-3} < x
\][/tex]
3. Multiply both sides by [tex]\( x-3 \)[/tex]:
Note that since [tex]\( x < 3 \)[/tex], [tex]\( x-3 \)[/tex] is negative, and multiplying or dividing both sides by a negative quantity reverses the inequality:
[tex]\[
\frac{4}{x-3} \cdot (x-3) > x \cdot (x-3)
\][/tex]
This simplifies to:
[tex]\[
4 > x(x-3)
\][/tex]
or
[tex]\[
4 > x^2 - 3x
\][/tex]
4. Rearrange the inequality:
Rewrite the inequality as a standard quadratic inequality:
[tex]\[
x^2 - 3x - 4 < 0
\][/tex]
5. Solve the quadratic inequality:
Solve the corresponding quadratic equation:
[tex]\[
x^2 - 3x - 4 = 0
\][/tex]
Factorize the quadratic equation:
[tex]\[
(x - 4)(x + 1) = 0
\][/tex]
The solutions to the equation are [tex]\( x = 4 \)[/tex] and [tex]\( x = -1 \)[/tex].
6. Determine the sign of the quadratic expression:
The roots [tex]\( x = 4 \)[/tex] and [tex]\( x = -1 \)[/tex] divide the x-axis into three intervals: [tex]\( (-\infty, -1) \)[/tex], [tex]\( (-1, 4) \)[/tex], and [tex]\( (4, \infty) \)[/tex]. To determine where the quadratic expression [tex]\( x^2 - 3x - 4 \)[/tex] is less than 0, test points from each interval:
- For [tex]\( x < -1 \)[/tex]: choose [tex]\( x = -2 \)[/tex].
[tex]\[
(-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6 \quad (\text{positive})
\][/tex]
- For [tex]\( -1 < x < 4 \)[/tex]: choose [tex]\( x = 0 \)[/tex].
[tex]\[
0^2 - 3(0) - 4 = -4 \quad (\text{negative})
\][/tex]
- For [tex]\( x > 4 \)[/tex]: choose [tex]\( x = 5 \)[/tex].
[tex]\[
5^2 - 3(5) - 4 = 25 - 15 - 4 = 6 \quad (\text{positive})
\][/tex]
Hence, the quadratic expression [tex]\( x^2 - 3x - 4 \)[/tex] is negative in the interval [tex]\((-1, 4)\)[/tex].
7. Incorporate the [tex]\( x < 3 \)[/tex] constraint:
Since we are interested only in values where [tex]\( x < 3 \)[/tex], intersect the interval [tex]\((-1, 4)\)[/tex] with [tex]\((- \infty, 3)\)[/tex]:
[tex]\[
(-1, 4) \cap (-\infty, 3) = (-1, 3)
\][/tex]
### Conclusion
The values of [tex]\( x \)[/tex] for which [tex]\( \frac{4}{x-3} + 2 < x + 2 \)[/tex] given [tex]\( x < 3 \)[/tex] are:
[tex]\[
-1 < x < 3
\][/tex]
