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2 years ago

Solve the following inequality.

1 answer:

7 0

First simplify.
$-\frac{2}{3}x \ \textgreater \ \mathrm{Multiply\:fractions}:\ \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c} \ \textgreater \ -\frac{2x}{3}$

$\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)} \ \textgreater \ \ \textgreater \ \frac{2x}{3}\ \textless \ -8$

$\mathrm{Multiply\:both\:sides\:by\:}3 \ \textgreater \ \frac{3\cdot \:2x}{3}\ \textless \ 3\left(-8\right) \ \textgreater \ Simplify \ \textgreater \ 2x\ \textless \ -24$

$\mathrm{Divide\:both\:sides\:by\:}2 \ \textgreater \ \frac{2x}{2}\ \textless \ \frac{-24}{2} \ \textgreater \ Simplify \ \textgreater \ x\ \textless \ -12$

$Therefore\; x\ \textless \ -12 \;is\;our\;answer!$

Hope this helps!

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The area enclosed by the graphs of y = 1/x, y = 1, and x = 3 is rotated about the line y = -1. Find the volume and show steps.

Natalija [7]

Answer:

$A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\approx9.8524$

Step-by-step explanation:

Use the Washer Method $A=\pi\displaystyle \int\limits_{a}^{b}{\bigr[R(x)^2-r(x)^2\bigr] \, dx$ where $R(x)$ is the outer radius and $r(x)$ is the inner radius.

If we sketch out the graph, we see that $y=1$ intersects points $(1,1)$ and $(3,1)$ , which will be our bounds of integration.

Here, our outer radius will be $R(x)=(-1-1)=-2$ and our inner radius will be $r(x)=-1-\frac{1}{x}$ .

Thus, we can compute the integral and find the volume:

$A=\pi\displaystyle\int\limits^{3}_{1} {(-2)^2-\biggr(-1-\frac{1}{x}\biggr)^2 } \, dx\\ \\A=\pi\displaystyle\int\limits^{3}_{1} {4-\biggr(1+\frac{2}{x}+\frac{1}{x^2} \biggr) } \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {4-1-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\int\limits^{3}_{1} {3-\frac{2}{x}-\frac{1}{x^2}} \, dx\\\\A=\pi\displaystyle\biggr[3x-2\ln(|x|)+\frac{1}{x}\biggr]\Biggr|_{1}^{3}\\$

$A=\pi\displaystyle\biggr[\biggr(3(3)-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3(1)-2\ln(|1|)+\frac{1}{1}\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(9-2\ln(|3|)+\frac{1}{3}\biggr)-\biggr(3+1\biggr)\biggr]\\\\A=\pi\displaystyle\biggr[\biggr(\frac{28}{3}-2\ln(|3|)\biggr)-\biggr(4\biggr)\biggr]\\A=\pi\displaystyle\biggr[\frac{16}{3}-2\ln(|3|)\biggr]\\A\approx9.8524$

In conclusion, the volume of the solid of revolution will be about 9.8524 cubic units. See the attached graph for a helpful visual!

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1 year ago

What’s the reciprocal of -8

denpristay [2]

The reciprocal of -8 will be -1/8

hope this helped.

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3 years ago

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Shtirlitz [24]

Answer:

Step-by-step explanation:

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2 years ago

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3 years ago

1. Write a two-column proof.

Alexandra [31]

Answer:

Check it below, please

Step-by-step explanation:

Hi there!

Let's prove segment AB is perpendicular to CD. Attention to the fact that a two column proof has to be concise. So all the comments can't be exhaustive, but as short as possible.

Let's recap: An isosceles triangle is one triangle with at least 2 congruent angles.

Statement Reason

$\overline{AB} \cong \overline{AC}\\$ Given

$\widehat{ACB} \:bisects\: \overline{AB}$ Isosceles Triangle the altitude, the bisector coincide.

$\overline{AD} \cong \overline{DB}$ Bisector equally divide a line segment into two congruent

$m\angle CDA =m \angle CBD=90^{\circ}$ Right angles, perpendicular lines.

$\overline{CD}\perp \overline{AB}$ Perpendicular Line segment

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3 years ago