Question

4.
a. $4(2x - 2) + 2 = 4(x - 2)$
b. $4x + 2(2x - 3) = 8(x - 1)$
c. $4x + 2(2x - 3) = 4(2x - 2) + 2$

Explanation:

Step1: Expand all parentheses

Part a:

$4(2x-2)+2=4(x-2)$
$8x-8+2=4x-8$

Part b:

$4x+2(2x-3)=8(x-1)$
$4x+4x-6=8x-8$

Part c:

$4x+2(2x-3)=4(2x-2)+2$
$4x+4x-6=8x-8+2$

Step2: Simplify both sides

Part a:

$8x-6=4x-8$

Part b:

$8x-6=8x-8$

Part c:

$8x-6=8x-6$

Step3: Isolate variable x (Part a)

Subtract $4x$ from both sides:
$8x-4x-6=4x-4x-8$
$4x-6=-8$
Add 6 to both sides:
$4x-6+6=-8+6$
$4x=-2$
Divide by 4:
$x=\frac{-2}{4}=-\frac{1}{2}$

Step4: Analyze for contradictions/identities

Part b:

Subtract $8x$ from both sides:
$8x-8x-6=8x-8x-8$
$-6=-8$ (contradiction, no solution)

Part c:

Subtract $8x$ from both sides:
$8x-8x-6=8x-8x-6$
$-6=-6$ (identity, infinite solutions)

Answer:

a. $x=-\frac{1}{2}$
b. No solution
c. All real numbers (infinite solutions)