To solve the inequalities \( 4x - 3 > 9 \) and \( x + 6 > -2 \), we will solve each inequality separately and then find their intersection.
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Solving \( 4x - 3 > 9 \): \[ 4x - 3 > 9 \] Add 3 to both sides: \[ 4x > 12 \] Now, divide both sides by 4: \[ x > 3 \]
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Solving \( x + 6 > -2 \): \[ x + 6 > -2 \] Subtract 6 from both sides: \[ x > -8 \]
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Finding the solution set: We now have two inequalities:
- \( x > 3 \)
- \( x > -8 \)
The more restrictive condition is \( x > 3 \), because any number greater than 3 will also satisfy \( x > -8 \).
So, the solution set for the given inequalities is: \[ \boxed{(3, \infty)} \]