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Sagot :
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]
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]
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