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PS · Problem Solving

605-655

Absolute Values/Modules

If $y = |8x - 24| + |-5x + 10| + |4x + 1|$ , at which x-value does y reach its minimum?

Answer Choices

Correct answer marked below

A

\frac{1}{2}

2

Correct

C

3

D

4

E

\frac{5}{2}

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Explanation

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Official Explanation

Official

Step 1: Identify Critical Points

Absolute value functions change their behavior at the points where their expressions inside the absolute values equal zero. We solve for these points:

8x - 24 = 0 \Rightarrow x = 3

-5x + 10 = 0 \Rightarrow x = 2

4x + 1 = 0 \Rightarrow x = -\frac{1}{4}

So, the critical points that divide the number line into intervals are

x = -\frac{1}{4}, 2, 3

.

Step 2: Analyze Each Interval

We evaluate

y

in each interval by considering the sign of each term.

Interval

(-\infty, -\frac{1}{4})

8x - 24 < 0 \Rightarrow |8x - 24| = -(8x - 24) = -8x + 24

-5x + 10 > 0 \Rightarrow |-5x + 10| = -5x + 10

4x + 1 < 0 \Rightarrow |4x + 1| = -(4x + 1) = -4x - 1

y = (-8x + 24) + (-5x + 10) + (-4x - 1) = -17x + 33

Interval

(-\frac{1}{4}, 2)

8x - 24 < 0 \Rightarrow |8x - 24| = -8x + 24

-5x + 10 > 0 \Rightarrow |-5x + 10| = -5x + 10

4x + 1 > 0 \Rightarrow |4x + 1| = 4x + 1

y = (-8x + 24) + (-5x + 10) + (4x + 1) = -9x + 35

Interval

(2,3)

8x - 24 < 0 \Rightarrow |8x - 24| = -8x + 24

-5x + 10 < 0 \Rightarrow |-5x + 10| = 5x - 10

…

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Comments

3

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T

TeddyMar 7, 2025

Thank for sharing!

K

KabitaApr 12, 2025

Any shortcuts for solving this faster?

P

pitrujaJul 14, 2024

thanks for posting

If $y = |8x - 24| + |-5x + 10| + |4x + 1|$ , at which x-value does y reach its minimum?

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If y=∣8x−24∣+∣−5x+10∣+∣4x+1∣y = |8x - 24| + |-5x + 10| + |4x + 1|y=∣8x−24∣+∣−5x+10∣+∣4x+1∣, at which x-value does y reach its minimum?

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Official Explanation

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What it tests

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If $y = |8x - 24| + |-5x + 10| + |4x + 1|$ , at which x-value does y reach its minimum?