MPT Case Analysis 4

Question 1

Using case analysis, find all solutions to |x + 2| = |x - 4|.

Accepted Answer

x = 1

Answer

Case 1: x + 2 = x - 4 gives -2 = -4 (impossible); Case 2: x + 2 = -(x - 4) gives 2x = 2, so x = 1.

Question 2

Analyze cases for the inequality |x - 3| < 2. What is the solution?

Accepted Answer

1 < x < 5

Answer

|x - 3| < 2 means -2 < x - 3 < 2; adding 3 throughout gives 1 < x < 5.

Question 3

A piecewise function is f(x) = x + 4 for x ≤ -2, f(x) = x² for -2 < x < 2, and f(x) = 4 - x for x ≥ 2. What is f(-2)?

Accepted Answer

2

Answer

Since x = -2 satisfies x ≤ -2, use f(x) = x + 4: f(-2) = -2 + 4 = 2.

Question 4

For x < -1, which expression equals |x + 1| - |x - 1|?

Accepted Answer

-2

Answer

For x < -1: |x+1| = -(x+1) and |x-1| = -(x-1); subtracting gives -(x+1) - (-(x-1)) = -x-1+x-1 = -2.

Question 5

A taxi charges a $3 base fare plus $2/mile for the first 5 miles and $1.50/mile after that. What is the cost for a 7-mile trip?

Accepted Answer

$16

Answer

First 5 miles: $3 + 5×$2 = $13; remaining 2 miles: 2×$1.50 = $3; total = $13 + $3 = $16.

(MPT) UCF Math Placement Test Practice Test