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Solve the equation. Be sure to check your proposed solution by substituting it for the variable in the original equation.

[tex]\[ 4(2x - 4) = 22 \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is [tex]\(\{\} \)[/tex]. (Type an integer or a simplified fraction.)
B. The solution set is [tex]\(\{x \mid x \text{ is a real number}\}\)[/tex].
C. The solution set is [tex]\(\varnothing\)[/tex].


Sagot :

Let's solve the equation step by step and verify our solution by substituting it back into the original equation.

Given equation:
[tex]\[ 4(2x - 4) = 22 \][/tex]

Step 1: Distribute the 4 on the left side of the equation.
[tex]\[ 4 \cdot (2x - 4) = 4 \cdot 2x - 4 \cdot 4 \][/tex]
[tex]\[ 8x - 16 = 22 \][/tex]

Step 2: To isolate the term containing [tex]\( x \)[/tex], add 16 to both sides of the equation.
[tex]\[ 8x - 16 + 16 = 22 + 16 \][/tex]
[tex]\[ 8x = 38 \][/tex]

Step 3: To solve for [tex]\( x \)[/tex], divide both sides of the equation by 8.
[tex]\[ x = \frac{38}{8} \][/tex]

Step 4: Simplify the fraction.
[tex]\[ x = \frac{19}{4} \][/tex]

So, the proposed solution is [tex]\( x = \frac{19}{4} \)[/tex].

Step 5: Verify the solution by substituting [tex]\( x = \frac{19}{4} \)[/tex] back into the original equation and check if both sides are equal.

Original equation:
[tex]\[ 4(2x - 4) = 22 \][/tex]

Substitute [tex]\( x = \frac{19}{4} \)[/tex]:
[tex]\[ 4 \left( 2 \cdot \frac{19}{4} - 4 \right) = 22 \][/tex]
[tex]\[ 4 \left( \frac{38}{4} - 4 \right) = 22 \][/tex]
[tex]\[ 4 \left( 9.5 - 4 \right) = 22 \][/tex]
[tex]\[ 4 \times 5.5 = 22 \][/tex]
[tex]\[ 22 = 22 \][/tex]

Since both sides of the equation are equal when [tex]\( x = \frac{19}{4} \)[/tex], the solution is verified.

Therefore, the solution set is:
[tex]\[ \left\{ \frac{19}{4} \right\} \][/tex]

So, the correct choice is:
A. The solution set is [tex]\( \left\{ \frac{19}{4} \right\} \)[/tex].