Question

1. $4 - \frac{3x - 1}{3} > \frac{x}{6} + \frac{7}{2}$
2. $|3 - 8x| < 19$
3. the perimeter of whitney’s rectangular driveway is 168 feet. the length is 12 feet longer than three times the width. what are the dimensions of the driveway?

Explanation:

Problem 4: Solve the inequality \( 4 - \frac{3x - 1}{3} > \frac{x}{6} + \frac{7}{2} \)

Step 1: Eliminate denominators

Multiply every term by 6 (the least common multiple of 3 and 6) to clear the fractions:
\( 6 \times 4 - 6 \times \frac{3x - 1}{3} > 6 \times \frac{x}{6} + 6 \times \frac{7}{2} \)
Simplify each term:
\( 24 - 2(3x - 1) > x + 21 \)

Step 2: Distribute and simplify

Distribute the -2 in \( -2(3x - 1) \):
\( 24 - 6x + 2 > x + 21 \)
Combine like terms:
\( 26 - 6x > x + 21 \)

Step 3: Isolate the variable

Add \( 6x \) to both sides:
\( 26 > 7x + 21 \)
Subtract 21 from both sides:
\( 5 > 7x \)

Step 4: Solve for \( x \)

Divide both sides by 7:
\( \frac{5}{7} > x \) or \( x < \frac{5}{7} \)

Problem 5: Solve the absolute value inequality \( |3 - 8x| < 19 \)

Step 1: Apply absolute value property

For \( |A| < B \) (where \( B > 0 \)), we have \( -B < A < B \). So:
\( -19 < 3 - 8x < 19 \)

Step 2: Subtract 3 from all parts

\( -19 - 3 < -8x < 19 - 3 \)
Simplify:
\( -22 < -8x < 16 \)

Step 3: Divide by -8 (and reverse inequalities)

Divide all parts by -8 (remember to reverse the inequality signs when dividing by a negative number):
\( \frac{-22}{-8} > x > \frac{16}{-8} \)
Simplify:
\( \frac{11}{4} > x > -2 \) or \( -2 < x < \frac{11}{4} \)

Problem 6: Find the dimensions of the rectangular driveway

Let \( w \) = width (in feet) and \( l \) = length (in feet).
Given:

• Perimeter \( P = 2(l + w) = 168 \)
• Length: \( l = 3w + 12 \)

Step 1: Substitute \( l \) into the perimeter formula

Substitute \( l = 3w + 12 \) into \( 2(l + w) = 168 \):
\( 2((3w + 12) + w) = 168 \)

Step 2: Simplify and solve for \( w \)

Simplify inside the parentheses:
\( 2(4w + 12) = 168 \)
Distribute the 2:
\( 8w + 24 = 168 \)
Subtract 24 from both sides:
\( 8w = 144 \)
Divide by 8:
\( w = 18 \)

Step 3: Find \( l \)

Substitute \( w = 18 \) into \( l = 3w + 12 \):
\( l = 3(18) + 12 = 54 + 12 = 66 \)

Answer:

s:

1. \( \boldsymbol{x < \frac{5}{7}} \)
2. \( \boldsymbol{-2 < x < \frac{11}{4}} \)
3. Width = \( \boldsymbol{18} \) feet, Length = \( \boldsymbol{66} \) feet