To determine the solution to the inequality [tex]\(\log(2x + 7) \geq -\frac{2}{3}x + 2\)[/tex], let's break it down step-by-step.
### Step 1: Identify the Domain
Firstly, to apply the logarithmic function [tex]\(\log(2x + 7)\)[/tex], the argument [tex]\(2x + 7\)[/tex] must be positive:
[tex]\[ 2x + 7 > 0 \][/tex]
[tex]\[ 2x > -7 \][/tex]
[tex]\[ x > -\frac{7}{2} \][/tex]
Thus, the domain of [tex]\(x\)[/tex] is [tex]\( x > -3.5 \)[/tex].
### Step 2: Find the Point of Equality
We need to determine the values of [tex]\(x\)[/tex] where the two expressions [tex]\(\log(2x + 7)\)[/tex] and [tex]\(-\frac{2}{3}x + 2\)[/tex] are equal.
[tex]\[
\log(2x + 7) = -\frac{2}{3}x + 2
\][/tex]
Solving this equation, we find:
[tex]\([x = 1.5]\)[/tex]
### Step 3: Analyze the Graphs and Inequality
Now that we know the domain [tex]\(x > -3.5\)[/tex] and the point [tex]\(x = 1.5\)[/tex] where the expressions are equal, let's analyze what happens around this point.
Since we are dealing with an inequality [tex]\(\log(2x + 7) \geq -\frac{2}{3}x + 2\)[/tex]:
- For [tex]\(x = 1.5\)[/tex], both sides of the inequality are equal.
- For [tex]\(x > 1.5\)[/tex], we need to determine if [tex]\(\log(2x + 7) \geq -\frac{2}{3}x + 2\)[/tex].
Given the provided results, we know [tex]\(\log(2x + 7) \geq -\frac{2}{3}x + 2\)[/tex] holds true for [tex]\(x \geq 1.5\)[/tex].
### Conclusion
Combining the domain [tex]\(x > -3.5\)[/tex] and our analysis, we conclude that the solution to the inequality [tex]\(\log(2x + 7) \geq -\frac{2}{3}x + 2\)[/tex] starts at [tex]\(x = 1.5\)[/tex] and continues to [tex]\(\infty\)[/tex].
Thus, the correct answer is:
[tex]\[ [1.5, \infty) \][/tex]
