Asked by Blake Churchill on Mar 10, 2024
Verified
Transform the absolute value inequality ∣2−2h∣≥7| 2 - 2 h | \geq 7∣2−2h∣≥7 into two separate inequalities.
A) 2−2h≤72 - 2 h \leq 72−2h≤7 or 2−2h≥−72 - 2 h \geq - 72−2h≥−7
B) 2−2h≥72 - 2 h \geq 72−2h≥7 or 2−2h≥72 - 2 h \geq 72−2h≥7
C) 2−2h≥72 - 2 h \geq 72−2h≥7 or 2−2h≥−72 - 2 h \geq - 72−2h≥−7
D) 2−2h≥−72 - 2 h \geq - 72−2h≥−7 or 2−2h≤−72 - 2 h \leq - 72−2h≤−7
E) 2−2h≥72 - 2 h \geq 72−2h≥7 or 2−2h≤−72 - 2 h \leq - 72−2h≤−7
Separate Inequalities
Involves dealing with two or more inequalities independently or in relation to each other.
Absolute Value Inequality
An inequality involving absolute values, which constrains the distance of a variable from zero on the real number line within certain bounds.
- Develop proficiency in solving absolute value inequalities.
Verified Answer
CT
corey tuckerMar 10, 2024
Final Answer :
E
Explanation :
To separate the absolute value inequality, we need to consider two cases, one where the expression inside the absolute value is positive and the other where it is negative.
When 2−2h2 - 2h2−2h is positive, we have:
2−2h≥72-2h\geq72−2h≥7
Solving this inequality, we get:
h≤−52h\leq- \frac{5}{2}h≤−25
When 2−2h2 - 2h2−2h is negative, we have:
−(2−2h)≥7-(2-2h)\geq7−(2−2h)≥7
Simplifying, we get:
2h−2≤−72h-2\leq-72h−2≤−7
Solving this inequality, we get:
h≤−52h\leq-\frac{5}{2}h≤−25
Thus, combining the two inequalities, we get:
h≤−52h\leq-\frac{5}{2}h≤−25 or 2−2h≥72-2h\geq72−2h≥7 or 2−2h≤−72-2h\leq-72−2h≤−7
The final inequality can be further simplified as:
h≥94h\geq\frac{9}{4}h≥49 or h≤−52h\leq-\frac{5}{2}h≤−25
When 2−2h2 - 2h2−2h is positive, we have:
2−2h≥72-2h\geq72−2h≥7
Solving this inequality, we get:
h≤−52h\leq- \frac{5}{2}h≤−25
When 2−2h2 - 2h2−2h is negative, we have:
−(2−2h)≥7-(2-2h)\geq7−(2−2h)≥7
Simplifying, we get:
2h−2≤−72h-2\leq-72h−2≤−7
Solving this inequality, we get:
h≤−52h\leq-\frac{5}{2}h≤−25
Thus, combining the two inequalities, we get:
h≤−52h\leq-\frac{5}{2}h≤−25 or 2−2h≥72-2h\geq72−2h≥7 or 2−2h≤−72-2h\leq-72−2h≤−7
The final inequality can be further simplified as:
h≥94h\geq\frac{9}{4}h≥49 or h≤−52h\leq-\frac{5}{2}h≤−25
Learning Objectives
- Develop proficiency in solving absolute value inequalities.