Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
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].
### 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].
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
