To solve for [tex]\( x \)[/tex] in the equation [tex]\( p = q \)[/tex] where [tex]\( p = \frac{12x - 4}{3} \)[/tex] and [tex]\( q = 2 \left(4x - \frac{4}{3} \right) \)[/tex], we follow these steps:
1. First, express both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in their simplified forms.
[tex]\[ p = \frac{12x - 4}{3} \][/tex]
[tex]\[ q = 2 \left(4x - \frac{4}{3} \right) \][/tex]
2. Distribute and simplify [tex]\( q \)[/tex]:
[tex]\[ q = 2 \left(4x - \frac{4}{3} \right) = 2 \cdot 4x - 2 \cdot \frac{4}{3} = 8x - \frac{8}{3} \][/tex]
So the equations are:
[tex]\[ p = \frac{12x - 4}{3} \][/tex]
[tex]\[ q = 8x - \frac{8}{3} \][/tex]
3. Set [tex]\( p \)[/tex] equal to [tex]\( q \)[/tex]:
[tex]\[ \frac{12x - 4}{3} = 8x - \frac{8}{3} \][/tex]
4. Clear the fractions by multiplying both sides by 3:
[tex]\[ 3 \cdot \frac{12x - 4}{3} = 3 \cdot \left(8x - \frac{8}{3}\right) \][/tex]
This simplifies to:
[tex]\[ 12x - 4 = 24x - 8 \][/tex]
5. Isolate [tex]\( x \)[/tex] by moving all terms involving [tex]\( x \)[/tex] to one side and the constants to the other:
[tex]\[ 12x - 24x = -8 + 4 \][/tex]
[tex]\[ -12x = -4 \][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-4}{-12} = \frac{4}{12} = \frac{1}{3} \][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{1}{3} \][/tex]
Or in decimal form:
[tex]\[ x \approx 0.333333333333333 \][/tex]
