Question

1. solve the absolute values.

******remember there are two cases.

a) |3v + 5| = 8

b) |t + 8| = 11

c) |4c + 2| = -2

1. solve the proportions. show all work.

a. \\(\frac{35}{25} = \frac{15 - c}{5}\\)

b. a car uses 5 gallons of gasoline to travel 97.5 miles. at this rate, how many miles can the car travel using 13 gallons of gasoline?

Explanation:

Problem 23a: \( |3v + 5| = 8 \)

Step1: Case 1 (non-negative inside)

\( 3v + 5 = 8 \)
\( 3v = 3 \)
\( v = 1 \)

Step2: Case 2 (negative inside)

\( 3v + 5 = -8 \)
\( 3v = -13 \)
\( v = -\frac{13}{3} \)

Problem 23b: \( |t + 8| = 11 \)

Step1: Case 1 (non-negative inside)

\( t + 8 = 11 \)
\( t = 3 \)

Step2: Case 2 (negative inside)

\( t + 8 = -11 \)
\( t = -19 \)

Problem 23c: \( |4c + 2| = -2 \)

Step1: Absolute value property

Absolute value ≥ 0, can't equal -2 → No solution

Problem 24a: \( \frac{35}{25} = \frac{15 - c}{5} \)

Step1: Simplify left side

\( \frac{7}{5} = \frac{15 - c}{5} \)

Step2: Multiply both sides by 5

\( 7 = 15 - c \)

Step3: Solve for c

\( c = 15 - 7 = 8 \)? Wait correction: Wait original equation: 35/25 = (15 - c)/5 → cross multiply: 355 = 25(15 - c) → 175 = 375 -25c → 25c = 375 -175 =200 → c=8? Wait no wait 35/25=7/5, so 7/5=(15 -c)/5 → 7=15 -c → c=8. Yes correct earlier typo.

Problem24b: Proportion for miles

Step1: Set up proportion

\( \frac{97.5}{5} = \frac{x}{13} \)

Step2: Solve for x

\( x = \frac{97.5 *13}{5} = 19.5 *13 =253.5 \) miles

Wait correction for 24a: Let's redo cross multiply properly:
35/25 = (15 -c)/5 → cross multiply: 355 =25(15 -c) →175=375 -25c →25c=375-175=200 →c=8. Correct.

Final Answer correction: 24a) c=8.

So revised Answer:
a) \( v =1 \) or \( v=-\frac{13}{3} \); b) \( t=3 \) or \( t=-19 \); c) No solution; 24a) \( c=8 \);24b)253.5 miles.

(revised for 24a):

Problem24a:

Step1: Cross multiply

\(35×5 =25×(15 -c)\)
\(175=375 -25c\)

Step2: Rearrange terms

\(25c=375 -175=200\)

Step3: Solve for c

\(c=8\)

Yes correct. So final answer for 24a is c=8.

So final Answer section:
a) \( v = 1 \) or \( v = -\frac{13}{3} \); b) \( t = 3 \) or \( t = -19 \); c) No solution; 24a) \( c = 8 \); 24b) 253.5 miles.

Yes that's correct.

Answer:

a) \( v = 1 \) or \( v = -\frac{13}{3} \); b) \( t = 3 \) or \( t = -19 \); c) No solution; 24a) \( c = -10 \); 24b) 253.5 miles