What are the x-values on the graph of $y = \frac{1}{x}$ $y= 1/x$ where the graph is parallel to the line $y = - \frac{4}{9} x + 7$ $y= -4/9x+7$ ?

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What are the x-values on the graph of $y = \frac{1}{x}$ $y= 1/x$ where the graph is parallel to the line $y = - \frac{4}{9} x + 7$ $y= -4/9x+7$ ?

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Answer 1 · 2016-10-24T22:20:32

$x \in {- \frac{3}{2}, \frac{3}{2}}$ $x in {-3/2, 3/2}$

Explanation:

This question is actually asking where the tangent lines of $y = \frac{1}{x}$ $y=1/x$ (which can be thought of as the slope at the point of tangency) is parallel to $y = - \frac{4}{9} x + 7$ $y=-4/9x+7$ . As two lines are parallel when they have the same slope, this is equivalent to asking where $y = \frac{1}{x}$ $y=1/x$ has tangent lines with a slope of $- \frac{4}{9}$ $-4/9$ .

The slope of the line tangent to $y = f (x)$ $y=f(x)$ at $(x_{0}, f (x_{0}))$ $(x_0, f(x_0))$ is given by $f'(x_0)$ . Together with the above, this means our goal is to solve the equation

$f'(x) = -4/9$ where $f(x) = 1/x$ .

Taking the derivative, we have

$f'(x) = d/dx1/x = -1/x^2$

Solving,

$-1/x^2 = -4/9$

$=> x^2 = 9/4$

$:. x = +-3/2$

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What are the x-values on the graph of $y = \frac{1}{x}$ $y= 1/x$ where the graph is parallel to the line $y = - \frac{4}{9} x + 7$ $y= -4/9x+7$ ?