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Solve for [tex]$w$[/tex].

[tex]\[ -\frac{7}{w-6} = \frac{1}{4w-24} + 1 \][/tex]

If there is more than one solution, separate them with commas. If there is no solution, select "No solution."

[tex]\[ w = \quad \boxed{} \][/tex]

[tex]\[ \boxed{\text{No solution}} \][/tex]

Sagot :

GinnyAnsweridn
Let's solve the equation:

[tex]\[ -\frac{7}{w-6} = \frac{1}{4w - 24} + 1 \][/tex]

First, let's simplify the right-hand side:

[tex]\[ \frac{1}{4w - 24} + 1 \][/tex]

We can observe that:

[tex]\[ 4w - 24 = 4(w - 6) \][/tex]

Therefore:

[tex]\[ \frac{1}{4w - 24} = \frac{1}{4(w - 6)} \][/tex]

So the original equation becomes:

[tex]\[ -\frac{7}{w-6} = \frac{1}{4(w-6)} + 1 \][/tex]

Next, let's find a common denominator for the terms on the right:

[tex]\[ \frac{1}{4(w-6)} + 1 = \frac{1}{4(w-6)} + \frac{4(w-6)}{4(w-6)} = \frac{1 + 4(w-6)}{4(w-6)} \][/tex]

Simplify the expression:

[tex]\[ \frac{1 + 4w - 24}{4(w-6)} = \frac{1 + 4w - 24}{4(w-6)} = \frac{4w - 23}{4(w-6)} \][/tex]

So the equation becomes:

[tex]\[ -\frac{7}{w-6} = \frac{4w - 23}{4(w-6)} \][/tex]

To clear the denominators, multiply both sides by [tex]\(4(w - 6)\)[/tex]:

[tex]\[ -7 \cdot 4 = 4w - 23 \][/tex]

Simplify and solve for [tex]\(w\)[/tex]:

[tex]\[ -28 = 4w - 23 \][/tex]

Add 23 to both sides:

[tex]\[ -28 + 23 = 4w \][/tex]

[tex]\[ -5 = 4w \][/tex]

Divide both sides by 4:

[tex]\[ w = -\frac{5}{4} \][/tex]

So the solution to the equation is:

[tex]\[ w = -\frac{5}{4} \][/tex]

[tex]\[ \boxed{-\frac{5}{4}} \][/tex]
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