A construction crew is lengthening a road. The crew is adding

Question

3
miles to the road each week. The road originally measured
62
miles long. The crew's goal is given by
≥+3x62110
, where
x
is the number of weeks the crew works.
Complete the parts below.
(a)Solve the given inequality and graph the solution on the number line below.

2
4
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16
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28
30
0
(b)Choose and complete the statement that best describes the solution.

Thecrew'sgoalistolengthentheroadtoatmostmiles.Thiswilltakeatleastweeks.
Thecrew'sgoalistolengthentheroadtoatleastmiles.Thiswilltakeatleastweeks.
Thecrew'sgoalistolengthentheroadtoatleastmiles.Thiswilltakeatmostweeks.

Answer 1

Let's solve the given inequality step by step.

The total length of the road after \( x \) weeks can be expressed as:

\[ 62 + 3x \]

The crew's goal is represented by the inequality:

\[ 62 + 3x \geq 110 \]

(a) Solve the inequality

1. Subtract 62 from both sides:

\[ 3x \geq 110 - 62 \]

\[ 3x \geq 48 \]

2. Divide both sides by 3:

\[ x \geq \frac{48}{3} \]

\[ x \geq 16 \]

Representing the solution on a number line

We need to graph the solution \( x \geq 16 \) on the provided number line. Here’s how the number line would look:

  2  4  6  8  10  12  14  16  18  20  22  24  26  28  30
  ---------------------------------------------------------
                           ↑
   [======================]

The solid line to the right of 16 indicates that the solution includes 16 and all numbers greater than 16.

(b) Complete the statement

From the information derived from the inequality \( x \geq 16 \):

• The crew's goal is to lengthen the road to at least 110 miles. This will take at least 16 weeks.

Thus, the correct completion of the statement is:

The crew's goal is to lengthen the road to at least 110 miles. This will take at least 16 weeks.