Question

for each equation, find a value for x that makes the equation true.
a. ( x div 3 = 12 )
d. ( -4x = -24 )
b. ( 2x + 3 = 20 )
e. ( 2(x - 4) = 10 )
c. ( \frac{4}{3}x = \frac{10}{3} )
f. ( -0.5x + 11 = -2 )

Explanation:

Part a: \( x \div 3 = 12 \)

Step1: Multiply both sides by 3

To solve for \( x \), we multiply both sides of the equation \( x \div 3 = 12 \) by 3. This is because dividing by 3 and multiplying by 3 are inverse operations, so they will cancel each other out on the left side.
\[
x \div 3 \times 3 = 12 \times 3
\]

Step2: Simplify both sides

Simplifying the left side, \( x \div 3 \times 3 \) just gives \( x \). On the right side, \( 12 \times 3 = 36 \).
\[
x = 36
\]

Step1: Subtract 3 from both sides

To isolate the term with \( x \), we subtract 3 from both sides of the equation \( 2x + 3 = 20 \). This will get rid of the +3 on the left side.
\[
2x + 3 - 3 = 20 - 3
\]

Step2: Simplify both sides

Simplifying the left side, \( 2x + 3 - 3 = 2x \). On the right side, \( 20 - 3 = 17 \). So we have \( 2x = 17 \).

Step3: Divide both sides by 2

To solve for \( x \), we divide both sides of the equation \( 2x = 17 \) by 2.
\[
x=\frac{17}{2}=8.5
\]

Step1: Multiply both sides by \( \frac{3}{4} \)

To solve for \( x \), we multiply both sides of the equation \( \frac{4}{3}x=\frac{10}{3} \) by the reciprocal of \( \frac{4}{3} \), which is \( \frac{3}{4} \). This will cancel out the \( \frac{4}{3} \) on the left side.
\[
\frac{3}{4}\times\frac{4}{3}x=\frac{3}{4}\times\frac{10}{3}
\]

Step2: Simplify both sides

Simplifying the left side, \( \frac{3}{4}\times\frac{4}{3}x = x \). On the right side, the 3s cancel out, and we have \( \frac{10}{4}=\frac{5}{2} = 2.5\).
\[
x=\frac{5}{2}
\]

Answer:

\( x = 36 \)

Part b: \( 2x + 3 = 20 \)