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:
Answer: Choice C [tex]a-7 \le 5[/tex] (a-7 is less than or equal to 5) and [tex]a-7 \ge -5[/tex] (a-7 is greater than or equal to -5)
[tex]|a-7| \le 5[/tex] is the same as [tex]-5 \le a-7 \le 5[/tex]
We can use the rule that if [tex]|x| \le k[/tex] then [tex]-k \le x \le k[/tex] where k is any positive number
After we get [tex]-5 \le a-7 \le 5[/tex], we break it down into two pieces. The first piece is [tex]-5 \le a-7 [/tex] while the second piece is [tex]a-7 \le 5[/tex]
Flip the two sides of [tex]-5 \le a-7 [/tex] to get [tex]a-7 \ge -5[/tex]
So we have [tex]a-7 \le 5[/tex] and [tex]a-7 \ge -5[/tex]
That's why the answer is choice C
Side Note: we must use the connector "and" instead of "or". Both parts of [tex]a-7 \le 5[/tex] and [tex]a-7 \ge -5[/tex] need to be true, instead of just one part only. This is why we can rule out choice B due to the "or" connector.