Question

part ii
answer all 4 questions in this part. each correct answer will receive 2 credits. clearly indicate the
necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. utilize the
information provided for each question to determine your answer. note that diagrams are not
necessarily drawn to scale. for all questions in this part, a correct numerical answer with no work
shown will receive only 1 credit.

1. solve for ( x ): ( \frac{1}{16}x + \frac{1}{4} = \frac{1}{2} )

1. solve algebraically for ( x ): ( 3(x + 1) - 5x = 12 - (6x - 7) )

Explanation:

Question 17

Step1: Subtract 1/4 from both sides

To isolate the term with \( x \), we subtract \( \frac{1}{4} \) from both sides of the equation \( \frac{1}{16}x+\frac{1}{4}=\frac{1}{2} \).

$$ \frac{1}{16}x+\frac{1}{4}-\frac{1}{4}=\frac{1}{2}-\frac{1}{4} $$

Simplifying the left - hand side (LHS) and right - hand side (RHS):
The LHS becomes \( \frac{1}{16}x \), and for the RHS, we find a common denominator (which is 4) for \( \frac{1}{2} \) and \( \frac{1}{4} \). So \( \frac{1}{2}=\frac{2}{4} \), then \( \frac{2}{4}-\frac{1}{4}=\frac{2 - 1}{4}=\frac{1}{4} \). So the equation is now \( \frac{1}{16}x=\frac{1}{4} \).

Step2: Multiply both sides by 16

To solve for \( x \), we multiply both sides of the equation \( \frac{1}{16}x=\frac{1}{4} \) by 16.

$$ 16\times\frac{1}{16}x = 16\times\frac{1}{4} $$

Simplifying both sides:
The LHS simplifies to \( x \), and the RHS: \( 16\times\frac{1}{4}=\frac{16}{4}=4 \).

Step1: Expand both sides of the equation

First, we expand the expressions on both sides of the equation \( 3(x + 1)-5x=12-(6x - 7) \).
For the left - hand side: Use the distributive property \( a(b + c)=ab+ac \), where \( a = 3 \), \( b=x \), and \( c = 1 \). So \( 3(x + 1)=3x+3 \). Then the left - hand side becomes \( 3x + 3-5x \). Combine like terms: \( 3x-5x=-2x \), so the LHS is \( -2x + 3 \).
For the right - hand side: Distribute the negative sign to \( 6x \) and \( - 7 \). So \( 12-(6x - 7)=12-6x + 7 \). Combine like terms: \( 12 + 7=19 \), so the RHS is \( -6x+19 \).
Now our equation is \( -2x + 3=-6x + 19 \).

Step2: Add 6x to both sides

To get all the \( x \) terms on one side, we add \( 6x \) to both sides of the equation \( -2x + 3=-6x + 19 \).

$$ -2x+6x + 3=-6x+6x + 19 $$

Simplifying both sides:
The LHS: \( -2x+6x = 4x \), so LHS is \( 4x + 3 \).
The RHS: \( -6x + 6x=0 \), so RHS is \( 19 \).
The equation is now \( 4x+3 = 19 \).

Step3: Subtract 3 from both sides

Subtract 3 from both sides of the equation \( 4x+3 = 19 \) to isolate the term with \( x \).

$$ 4x+3-3=19 - 3 $$

Simplifying both sides:
LHS: \( 4x+3 - 3=4x \)
RHS: \( 19-3 = 16 \)
The equation is now \( 4x=16 \).

Step4: Divide both sides by 4

Divide both sides of the equation \( 4x = 16 \) by 4.

$$ \frac{4x}{4}=\frac{16}{4} $$

Simplifying both sides:
LHS: \( \frac{4x}{4}=x \)
RHS: \( \frac{16}{4}=4 \)

Answer:

\( x = 4 \)

Question 18