To determine the solution set for the inequality \(\log (2x + 7) \geq -\frac{2}{3}x + 2\), we can follow these steps:
### Step 1: Consider the domain of the logarithmic function.
First, understand that the function \(\log(2x + 7)\) is only defined when the argument inside the logarithm is positive:
[tex]\[ 2x + 7 > 0 \][/tex]
[tex]\[ x > -\frac{7}{2} \][/tex]
Thus, \( x > -3.5 \).
### Step 2: Analyze the function and related inequalities.
Let's rewrite the inequality for clarity:
[tex]\[ \log(2x + 7) \geq -\frac{2}{3}x + 2 \][/tex]
### Step 3: Solve for the critical points.
To understand the solution better, let's find the point where the two expressions are equal:
[tex]\[ \log(2x + 7) = -\frac{2}{3}x + 2 \][/tex]
This is typically challenging to solve algebraically due to the logarithmic function and the linear function on the other side. Instead, we can analyze the intersection points graphically or through numerical methods.
### Step 4: Examine the intersection graphically.
Imagine plotting two functions:
1. \( f(x) = \log(2x + 7) \)
2. \( g(x) = -\frac{2}{3}x + 2 \)
We need to find where \( f(x) \geq g(x) \).
### Step 5: Determine the intersection points.
By solving these analytically or graphically, we find that the functions intersect approximately at \( x \approx 1.5 \).
### Step 6: Validate the intervals.
- For \( x < -3.5 \), the logarithmic function is undefined, so those values are not in the solution set.
- For \(-3.5 < x < 1.5 \), the linear equation \( g(x) \) is above or equal to the logarithmic curve \( f(x) \).
- For \( x \geq 1.5 \), the logarithmic curve \( f(x) \) is above the linear equation \( g(x) \).
### Step 7: Final solution interval.
The solution to \(\log(2x + 7) \geq -\frac{2}{3}x + 2\) includes values where the logarithmic function is greater than or equal to the linear function. These values \( x \) lie in the interval:
[tex]\[ [1.5, \infty) \][/tex]
Therefore, the set showing the solution to \(\log(2x + 7) \geq -\frac{2}{3}x + 2\) is:
[tex]\[ \boxed{[1.5, \infty)} \][/tex]
