Let's solve the inequalities step-by-step and match each one with the correct solution.
1. Inequality: [tex]\(4x - 3 > 5\)[/tex] and [tex]\(6x + 2 < -10\)[/tex]
- Solve [tex]\(4x - 3 > 5\)[/tex]:
[tex]\[
4x - 3 > 5 \\
4x > 8 \\
x > 2
\][/tex]
- Solve [tex]\(6x + 2 < -10\)[/tex]:
[tex]\[
6x + 2 < -10 \\
6x < -12 \\
x < -2
\][/tex]
Since [tex]\(x\)[/tex] cannot be both greater than 2 and less than -2 at the same time, there is no solution.
2. Inequality: [tex]\(|2x + 4| < 2\)[/tex]
- Rewrite the absolute value inequality as two separate inequalities:
[tex]\[
-2 < 2x + 4 < 2
\][/tex]
- Solve the compound inequality:
[tex]\[
-2 < 2x + 4 \\
-2 - 4 < 2x \\
-6 < 2x \\
-3 < x
\][/tex]
and
[tex]\[
2x + 4 < 2 \\
2x < 2 - 4 \\
2x < -2 \\
x < -1
\][/tex]
Putting these together, the solution for [tex]\(|2x + 4| < 2\)[/tex] is:
[tex]\[
-3 < x < -1
\][/tex]
3. Inequality: [tex]\(|x - 2| + 5 < 4\)[/tex]
- Isolate the absolute value term:
[tex]\[
|x - 2| + 5 < 4 \\
|x - 2| < -1
\][/tex]
Since the absolute value expression cannot be less than a negative number, there is no solution.
4. Inequality: [tex]\(|3x| - 3 < 3\)[/tex]
- Isolate the absolute value term:
[tex]\[
|3x| - 3 < 3 \\
|3x| < 6
\][/tex]
- Rewrite the absolute value inequality as two separate inequalities:
[tex]\[
-6 < 3x < 6
\][/tex]
- Solve the compound inequality:
[tex]\[
-6 < 3x \\
\frac{-6}{3} < x \\
-2 < x
\][/tex]
and
[tex]\[
3x < 6 \\
x < \frac{6}{3} \\
x < 2
\][/tex]
Putting these together, the solution for [tex]\(|3x| - 3 < 3\)[/tex] is:
[tex]\[
-2 < x < 2
\][/tex]
Now we can clearly match the inequalities to their solutions:
- [tex]\(4x - 3 > 5\)[/tex] and [tex]\(6x + 2 < -10\)[/tex]: no solution
- [tex]\(|2x + 4| < 2\)[/tex]: [tex]\(-3 < x < -1\)[/tex]
- [tex]\(|x - 2| + 5 < 4\)[/tex]: no solution
- [tex]\(|3x| - 3 < 3\)[/tex]: [tex]\(-2 < x < 2\)[/tex]
