To solve the inequalities [tex]\( 2z \leq 3z - 7 \)[/tex] or [tex]\( 3 - 4z > 31 \)[/tex], we need to analyze each inequality separately and then combine the results.
### Inequality 1: [tex]\( 2z \leq 3z - 7 \)[/tex]
1. Start by isolating [tex]\( z \)[/tex] on one side of the inequality:
[tex]\[
2z \leq 3z - 7
\][/tex]
2. Subtract [tex]\( 3z \)[/tex] from both sides to get:
[tex]\[
2z - 3z \leq -7
\][/tex]
3. Simplify the left side:
[tex]\[
-z \leq -7
\][/tex]
4. To isolate [tex]\( z \)[/tex], divide both sides by [tex]\(-1\)[/tex]. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
z \geq 7
\][/tex]
The solution to the first inequality is:
[tex]\[ z \geq 7 \][/tex]
### Inequality 2: [tex]\( 3 - 4z > 31 \)[/tex]
1. Start by isolating [tex]\( z \)[/tex] on one side of the inequality:
[tex]\[
3 - 4z > 31
\][/tex]
2. Subtract 3 from both sides to get:
[tex]\[
-4z > 28
\][/tex]
3. Divide both sides by [tex]\(-4\)[/tex] and remember to reverse the inequality sign:
[tex]\[
z < -7
\][/tex]
The solution to the second inequality is:
[tex]\[ z < -7 \][/tex]
### Combining the Results
We need to find the union of both inequalities, because the problem states [tex]\( 2z \leq 3z - 7 \)[/tex] or [tex]\( 3 - 4z > 31 \)[/tex].
For the combined solution, we have:
- From the first inequality: [tex]\( z \geq 7 \)[/tex]
- From the second inequality: [tex]\( z < -7 \)[/tex]
### Final Solution
- The solution for [tex]\( 2z \leq 3z - 7 \)[/tex] is [tex]\( z \geq 7 \)[/tex], which is written as the interval: [tex]\([7, +\infty)\)[/tex]
- The solution for [tex]\( 3 - 4z > 31 \)[/tex] is [tex]\( z < -7 \)[/tex], which is written as the interval: [tex]\((-\infty, -7)\)[/tex]
So, the complete solution for the inequality [tex]\( 2z \leq 3z - 7 \)[/tex] or [tex]\( 3 - 4z > 31 \)[/tex] is in interval notation:
[tex]\[ (-\infty, -7) \cup [7, +\infty) \][/tex]
The result means [tex]\( z \)[/tex] belongs to either [tex]\( (-\infty, -7) \)[/tex] or [tex]\([7, +\infty) \)[/tex].
