Find the solution set of $4x-3\gt 9$ and $x+6\gt -2$

Question

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: \[ \boxed{D { x \mid x > 3 }} \]