Let's solve the given system of inequalities step-by-step.
### Solving the first inequality: [tex]\( 4x + 2 > 10 \)[/tex]
1. Subtract 2 from both sides:
[tex]\[
4x + 2 - 2 > 10 - 2
\][/tex]
This simplifies to:
[tex]\[
4x > 8
\][/tex]
2. Divide both sides by 4:
[tex]\[
\frac{4x}{4} > \frac{8}{4}
\][/tex]
This simplifies to:
[tex]\[
x > 2
\][/tex]
Thus, the solution to the first inequality is [tex]\( x > 2 \)[/tex].
### Solving the second inequality: [tex]\( -3x - 1 > 5 \)[/tex]
1. Add 1 to both sides:
[tex]\[
-3x - 1 + 1 > 5 + 1
\][/tex]
This simplifies to:
[tex]\[
-3x > 6
\][/tex]
2. Divide both sides by -3, and importantly, reverse the inequality sign (since we are dividing by a negative number):
[tex]\[
\frac{-3x}{-3} < \frac{6}{-3}
\][/tex]
This simplifies to:
[tex]\[
x < -2
\][/tex]
Thus, the solution to the second inequality is [tex]\( x < -2 \)[/tex].
### Examining the overall solution
Now we combine the two solutions:
- From the first inequality, we found [tex]\( x > 2 \)[/tex].
- From the second inequality, we found [tex]\( x < -2 \)[/tex].
We need to find the region where both inequalities are true simultaneously. However, there is no value of [tex]\( x \)[/tex] that can satisfy both [tex]\( x > 2 \)[/tex] and [tex]\( x < -2 \)[/tex] at the same time. These regions do not overlap.
### Conclusion
Since there is no overlap between [tex]\( x > 2 \)[/tex] and [tex]\( x < -2 \)[/tex], there is no solution that satisfies both inequalities simultaneously. Therefore, the final answer is:
[tex]\[
\text{No solution since there is no overlap between } x > 2 \text{ and } x < -2.
\][/tex]
