Sure! Let's match each inequality with its correct solution.
1. [tex]\(4x + 2 > 10\)[/tex] and [tex]\(-3x - 1 > 5\)[/tex]
- Solving the first part:
[tex]\[
4x + 2 > 10 \implies 4x > 8 \implies x > 2
\][/tex]
- Solving the second part:
[tex]\[
-3x - 1 > 5 \implies -3x > 6 \implies x < -2
\][/tex]
- Combining the solutions, we see that [tex]\(x > 2\)[/tex] and [tex]\(x < -2\)[/tex] cannot be simultaneously true.
Therefore, the solution is "no solution".
2. [tex]\(|3x| + 4 < 10\)[/tex]
- Simplifying:
[tex]\[
|3x| < 6
\][/tex]
- This translates to:
[tex]\[
-6 < 3x < 6 \implies -2 < x < 2
\][/tex]
Therefore, the solution is [tex]\(-2 < x < 2\)[/tex].
3. [tex]\(|x+2| + 4 < 3\)[/tex]
- Simplifying:
[tex]\[
|x + 2| < -1
\][/tex]
- Since an absolute value cannot be less than a negative number, this inequality has no solution.
Therefore, the solution is "no solution".
4. [tex]\(|2x+4| + 2 > 4\)[/tex]
- Simplifying:
[tex]\[
|2x + 4| > 2
\][/tex]
- This translates to two separate inequalities:
[tex]\[
2x + 4 > 2 \implies 2x > -2 \implies x > -1
\][/tex]
[tex]\[
2x + 4 < -2 \implies 2x < -6 \implies x < -3
\][/tex]
- Therefore, the solution is:
[tex]\(x > -1 \text{ or } x < -3\)[/tex]
In summary, we have:
1. [tex]\(4x + 2 > 10\)[/tex] and [tex]\(-3x - 1 > 5\)[/tex] corresponds to "no solution".
2. [tex]\(|3x| + 4 < 10\)[/tex] corresponds to "-2 < x < 2".
3. [tex]\(|x+2| + 4 < 3\)[/tex] corresponds to "no solution".
4. [tex]\(|2x + 4| + 2 > 4\)[/tex] corresponds to "x > -1 \text{ or } x < -3".
