Question

Find the solution set of $4x-3\gt 9$ and $x+6\gt -2$
The solution set is null
$\{ 3,-8\} $
D $\{ x\vert x\gt 3\} $
$\{ x\vert x\gt -8\} $

Answer 1

To solve the inequalities \(4x - 3 > 9\) and \(x + 6 > -2\), we will solve each inequality separately.

1. **Solving \(4x - 3 > 9\)**:
\[
4x - 3 > 9
\]
First, add 3 to both sides:
\[
4x > 9 + 3
\]
\[
4x > 12
\]
Next, divide both sides by 4:
\[
x > 3
\]

2. **Solving \(x + 6 > -2\)**:
\[
x + 6 > -2
\]
First, subtract 6 from both sides:
\[
x > -2 - 6
\]
\[
x > -8
\]

Now we have the solutions to both inequalities:
- From the first inequality, we have \(x > 3\).
- From the second inequality, we have \(x > -8\).

Next, we need to find the intersection of these two solution sets.
- The solution set \(x > 3\) contains all numbers greater than 3.
- The solution set \(x > -8\) contains all numbers greater than -8.

Since \(x > 3\) is a stricter condition than \(x > -8\) (every number that is greater than 3 is also greater than -8), the intersection of the two sets is simply:
\[
x > 3
\]

Thus, the final solution set is:
\[
\{ x \mid x > 3 \}
\]

So, the answer from the given options is:
\[
\boxed{D \{ x \mid x > 3 \}}
\]