gloryyy3
Answered

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A rational expression has been simplified below.

[tex]\[
\frac{(x-2)(x+6)}{7(x+6)}=\frac{x-2}{7}
\][/tex]

For what values of [tex]\( x \)[/tex] are the two expressions equal?

A. All real numbers except 2
B. All real numbers except -6
C. All real numbers except -6 and 2
D. All real numbers

Sagot :

GinnyAnswer
Let's solve this step-by-step.

Given the rational expression:

[tex]\[ \frac{(x-2)(x+6)}{7(x+6)} = \frac{x-2}{7} \][/tex]

We want to find the values of [tex]\( x \)[/tex] for which these expressions are equal.

1. Simplify the Expression:

We notice that (x+6) is a common factor in the numerator and the denominator of the left side of the equation. Thus, we can simplify the expression by canceling out the common factor (x+6), given that [tex]\( x \neq -6 \)[/tex] (since division by zero is undefined).

So,
[tex]\[ \frac{(x-2)(x+6)}{7(x+6)} = \frac{x-2}{7} \quad \text{for} \quad x \neq -6. \][/tex]

2. Verify Simplification:

After canceling, we have:
[tex]\[ \frac{x-2}{7}. \][/tex]
Now both sides of the equation are equal:
[tex]\[ \frac{x-2}{7} = \frac{x-2}{7}. \][/tex]

These expressions are indeed equal for all [tex]\( x \)[/tex].

3. Identify Restrictions:

The only restriction comes from the cancellation step. The denominator [tex]\( x+6 \)[/tex] must not be zero, as division by zero is not allowed.

Therefore, setting the denominator equal to zero and solving for [tex]\( x \)[/tex] gives:
[tex]\[ x + 6 = 0 \implies x = -6. \][/tex]

So, [tex]\( x = -6 \)[/tex] is the value that makes the denominator zero and thus is not allowed.

Therefore, the two expressions are equal for all real numbers except [tex]\( x = -6 \)[/tex].

Thus, the correct answer is:

[tex]\[ \boxed{\text{B. All real numbers except -6}} \][/tex]
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