Question

1. Which conjunction or disjunction is equivalent to the open sentence |a – 8| ≤ 3?. A. a – 8 ≤ 3 and a – 8 ≥ –3. B. a – 8 ≥ 3 and a – 8 ≤ –3. C. a – 8 ≤ 3 or a – 8 ≥ –3. D. a – 8 ≥ 3 or a – 8 ≤ –3. 2. Which conjunction or disjunction is equivalent to the open sentence 4 – 2|n + 6| ≥ 2?. A. n + 6 ≤ –1 and n + 6 ≥ 1. B. –5 < n or n < –7. C. n + 6 ≤ 1 and n + 6 ≥ –1. D. n > –5 or n < –7. 3. Solve the equation. |k + 6| = 3. A. {–3, 9}. B. {–3, 3}. C. {–9, –3}. D. {all real numbers greater than or equal to –9 and less than or equal to –3}

Answer 1

The correct answers are:

A. a – 8 ≤ 3 and a – 8 ≥ –3; C. n + 6 ≤ 1 and n + 6 ≥ –1.; and C. {–9, –3}

Explanation:

Since absolute value is the distance from 0, when solving an absolute value equation or inequality, we must consider both positive and negative answers.

For the first question, |a-8|≤3, we break this into two inequalities, one with positive 3 at the end and one with negative 3 at the end.  However, with an inequality, if we change the sign of the answer, we must flip the inequality sign; this gives us

a-8≤3 and a-8≥-3.

(It is "and" since it was a less than or equal to inequality.)

For the second question, 4-2|n+6|≥2, we start out cancelling the terms outside the absolute value bars.  We begin by subtracting 4 from each side:

4-2|n+6|-4≥2-4

-2|n+6|≥-2

Now we divide both sides by -2.  Remember when you divide an inequality by a negative number, you flip the symbol:

(-2|n+6|)/-2 ≥ -2/-2

|n+6| ≤ 1

Now we split it into two inequalities.  Since it is "less than or equal to," they will be joined by "and":

n+6≤1 and n+6≥-1

For the last question, |k+6|=3, we split it into two equations:

k+6=3 or k+6=-3 (It is "or" because you cannot be equal to two numbers at the same time)

Solving each by subtracting 6 from each side:

k+6-6=3-6 or k+6-6=-3-6

k=-3 or k=-9

This gives us the set {-9, -3} for our solution.

Answer 2

The given expressions above are examples of an absolute-value inequality. In the expression |a – 8| ≤ 3, this is equal to -3 ≤ a - 8 ≤ 3. In this case, the answer here in 1 is A. a – 8 ≤ 3 and a – 8 ≥ –3. in 2, the expression is 4 – 2(n + 6) ≥ 2 or -8 -2n ≥ 2 or 4 + n ≤ -1 equal to n ≤ -5. The other equation is  4 – 2*-(n + 6) ≥ 2. 16 + 2n ≥ 2 or 8 + n ≥ 1. n ≥ -7.  Answer is n ≤ -5 or  n ≥ -7 .The equation |k + 6| = 3 is equal to k + 6 = 3 or k =-3 and k+6 =-3 equal to k = -9. Answer in 3 is C.