Question

Show that the equation 1/60 = 2x+50/x(x+50) is equivalent to x²-70x-3,000 = 0 for all values of x

not equal to 0 or -50. Explain each step as you rewrite the original equation.

Answer 1

See below.

Explanation:

Given equation:

`(1)/(60)=(2x+50)/(x(x+50))`

Multiply both sides by 60:

`\implies 60 \cdot \frac {1}{60}=60 \cdot (2x+50)/(x(x+50))`

`\implies 1=(60(2x+50))/(x(x+50))`

Multiply both sides by x(x+50):

`\implies 1\cdot x(x+50)=(60(2x+50))/(x(x+50)) \cdot x(x+50)`

`\implies x(x+50)=60(2x+50)`

Distribute both sides:

`\implies x \cdot x + x \cdot 50=60 \cdot 2x+60 \cdot 50`

`\implies x^2+50x=120x+3000`

Subtract 120x from both sides:

`\implies x^2+50x-120x=120x+3000-120x`

`\implies x^2-70x=3000`

Subtract 3000 from both sides:

`\implies x^2-70x-3000=3000-3000`

`\implies x^2-70x-3000=0`

A rational function is undefined when its denominator is equal to zero.

Therefore, to find the excluded x-values of the domain, set the denominator of the original function to zero and solve for x:

`\implies x(x+50)=0`

`\implies x=0`

`\implies x+50=0 \implies x=-50`

Thus proving that

`(1)/(60)=(2x+50)/(x(x+50))\;\;\; \textsf{is equivalent to}\;\;\;x^2-70x-3000=0`

for all values of x not equal to zero or -50.