Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Certainly! Let's solve these questions one by one.
### Question 10
We are given the equation [tex]\( |3 - 2x| = |5x + 2| + 1 \)[/tex] and asked to find the solution set.
From our solution process, we find that the equation has two distinct solutions:
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = 0 \)[/tex]
Hence, we can compare these solutions with each of the given choices:
A. [tex]\(\left\{-2, -\frac{4}{3}, 0, \frac{2}{7}\right\}\)[/tex] - This set does not match because it contains extraneous elements [tex]\(-2\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
B. [tex]\(\left\{-\frac{4}{3}, 0\right\}\)[/tex] - This is exactly the set of solutions we found.
C. [tex]\(\left\{-\frac{4}{3}, 0, \frac{2}{7}\right\}\)[/tex] - This set has an extraneous element [tex]\(\frac{2}{7}\)[/tex].
D. [tex]\(\{0\}\)[/tex] - This set only contains one of our solutions.
Thus, the correct answer is:
B. [tex]\(\left\{-\frac{4}{3}, 0\right\}\)[/tex]
### Question 11
We need to solve the inequality [tex]\(\left|\frac{2}{3} - 2x\right| \geq 2\)[/tex].
To handle this absolute value inequality, we break it into two separate inequalities based on the definition of absolute value:
1. [tex]\(\frac{2}{3} - 2x \geq 2\)[/tex]:
[tex]\[ \frac{2}{3} - 2x \geq 2 \implies -2x \geq 2 - \frac{2}{3} \implies -2x \geq \frac{6}{3} - \frac{2}{3} \implies -2x \geq \frac{4}{3} \][/tex]
[tex]\[ x \leq -\frac{4}{6} \implies x \leq -\frac{2}{3} \][/tex]
2. [tex]\(\frac{2}{3} - 2x \leq -2\)[/tex]:
[tex]\[ \frac{2}{3} - 2x \leq -2 \implies -2x \leq -2 - \frac{2}{3} \implies -2x \leq -\frac{6}{3} - \frac{2}{3} \implies -2x \leq -\frac{8}{3} \][/tex]
[tex]\[ x \geq \frac{8}{6} \implies x \geq \frac{4}{3} \][/tex]
Combining these two inequalities, we get:
[tex]\[ x \leq -\frac{2}{3} \text{ or } x \geq \frac{4}{3} \][/tex]
Let's compare these intervals against each of the given choices:
A. [tex]\(\left(-\infty, \frac{1}{6}\right) \cup \left(\frac{13}{6}, \infty\right)\)[/tex] - This does not match our intervals.
B. [tex]\(\left(-\infty, \frac{1}{6}\right) \cup \left[\frac{13}{6}, \infty\right)\)[/tex] - This does not match either.
C. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left(\frac{13}{6}, \infty\right)\)[/tex] - This does not include the boundary points we have.
D. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left[\frac{13}{6}, \infty\right)\)[/tex] - This does not match because of the boundaries provided.
So, correcting for the intervals, the closest correct intervals are:
C. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left(\frac{13}{6}, \infty\right)\)[/tex]
Our correction:
Answer should be:
[tex]\(\left(-\infty, -\frac{2}{3}\right] \cup \left[\frac{4}{3}, \infty\right)\)[/tex] which should match the intervals in choices correctly
### Question 10
We are given the equation [tex]\( |3 - 2x| = |5x + 2| + 1 \)[/tex] and asked to find the solution set.
From our solution process, we find that the equation has two distinct solutions:
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = 0 \)[/tex]
Hence, we can compare these solutions with each of the given choices:
A. [tex]\(\left\{-2, -\frac{4}{3}, 0, \frac{2}{7}\right\}\)[/tex] - This set does not match because it contains extraneous elements [tex]\(-2\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
B. [tex]\(\left\{-\frac{4}{3}, 0\right\}\)[/tex] - This is exactly the set of solutions we found.
C. [tex]\(\left\{-\frac{4}{3}, 0, \frac{2}{7}\right\}\)[/tex] - This set has an extraneous element [tex]\(\frac{2}{7}\)[/tex].
D. [tex]\(\{0\}\)[/tex] - This set only contains one of our solutions.
Thus, the correct answer is:
B. [tex]\(\left\{-\frac{4}{3}, 0\right\}\)[/tex]
### Question 11
We need to solve the inequality [tex]\(\left|\frac{2}{3} - 2x\right| \geq 2\)[/tex].
To handle this absolute value inequality, we break it into two separate inequalities based on the definition of absolute value:
1. [tex]\(\frac{2}{3} - 2x \geq 2\)[/tex]:
[tex]\[ \frac{2}{3} - 2x \geq 2 \implies -2x \geq 2 - \frac{2}{3} \implies -2x \geq \frac{6}{3} - \frac{2}{3} \implies -2x \geq \frac{4}{3} \][/tex]
[tex]\[ x \leq -\frac{4}{6} \implies x \leq -\frac{2}{3} \][/tex]
2. [tex]\(\frac{2}{3} - 2x \leq -2\)[/tex]:
[tex]\[ \frac{2}{3} - 2x \leq -2 \implies -2x \leq -2 - \frac{2}{3} \implies -2x \leq -\frac{6}{3} - \frac{2}{3} \implies -2x \leq -\frac{8}{3} \][/tex]
[tex]\[ x \geq \frac{8}{6} \implies x \geq \frac{4}{3} \][/tex]
Combining these two inequalities, we get:
[tex]\[ x \leq -\frac{2}{3} \text{ or } x \geq \frac{4}{3} \][/tex]
Let's compare these intervals against each of the given choices:
A. [tex]\(\left(-\infty, \frac{1}{6}\right) \cup \left(\frac{13}{6}, \infty\right)\)[/tex] - This does not match our intervals.
B. [tex]\(\left(-\infty, \frac{1}{6}\right) \cup \left[\frac{13}{6}, \infty\right)\)[/tex] - This does not match either.
C. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left(\frac{13}{6}, \infty\right)\)[/tex] - This does not include the boundary points we have.
D. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left[\frac{13}{6}, \infty\right)\)[/tex] - This does not match because of the boundaries provided.
So, correcting for the intervals, the closest correct intervals are:
C. [tex]\(\left(-\infty, \frac{1}{6}\right] \cup \left(\frac{13}{6}, \infty\right)\)[/tex]
Our correction:
Answer should be:
[tex]\(\left(-\infty, -\frac{2}{3}\right] \cup \left[\frac{4}{3}, \infty\right)\)[/tex] which should match the intervals in choices correctly
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.