Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly 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
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.