Q:

# Which conjunction or disjunction is equivalent to the open sentence |a – 7| ≤ 5?a – 7 ≥ 5 or a – 7 ≤ – 5a – 7 ≤ 5 or a – 7 ≥ – 5a – 7 ≤ 5 and a – 7 ≥ – 5a – 7 ≥ 5 and a – 7 ≤ – 5

Accepted Solution

A:
$$a-7 \le 5$$ (a-7 is less than or equal to 5)
and
$$a-7 \ge -5$$ (a-7 is greater than or equal to -5)

----------------------------------------------------

Explanation:

$$|a-7| \le 5$$ is the same as $$-5 \le a-7 \le 5$$

We can use the rule that if $$|x| \le k$$ then $$-k \le x \le k$$ where k is any positive number

After we get $$-5 \le a-7 \le 5$$, we break it down into two pieces. The first piece is $$-5 \le a-7$$ while the second piece is $$a-7 \le 5$$

Flip the two sides of $$-5 \le a-7$$ to get $$a-7 \ge -5$$

So we have $$a-7 \le 5$$ and $$a-7 \ge -5$$

That's why the answer is choice C

Side Note: we must use the connector "and" instead of "or". Both parts of $$a-7 \le 5$$ and $$a-7 \ge -5$$ need to be true, instead of just one part only. This is why we can rule out choice B due to the "or" connector.