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Elden [556K]
3 years ago
11

When is the equation

Formula1" title="\frac{x-7}{x^2-4x-21} =\frac{1}{x+3}" alt="\frac{x-7}{x^2-4x-21} =\frac{1}{x+3}" align="absmiddle" class="latex-formula"> true?
Mathematics
1 answer:
Vladimir79 [104]3 years ago
8 0

Notice that

x^2 - 4x - 21 = (x - 7) (x + 3)

so on the left side, a factor of <em>x</em> - 7 in the numerator and denominator can cancel

\dfrac{x-7}{x^2-4x-21} = \dfrac{x-7}{(x-7)(x+3)} = \dfrac1{x+3}

But we can only cancel them out as long as <em>x</em> ≠ 7 (because otherwise we'd have the indeterminate form 0/0 on the left side).

So, the equation is true for all <em>x</em> <u>except</u> <em>x</em> = 7. In set notation,

\{x \in \mathbb R \mid x \neq 7\}

In interval notation,

(-\infty,7)\cup(7,\infty)

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