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2 years ago

For the following exercises, solve the inequality and express the solution using interval notation.

Mathematics

1 answer:

lara [203]2 years ago

3 0

Answer:

In interval notation, the solution of |3 x-2|<7 is $-\frac{5}{3} < x < 3$ or $\left\{-\frac{5}{3}, 3\right\}$ .

Step-by-step explanation:

The given absolute value function is |3 x-2|<7.

It is required to solve the inequality and express the solution using interval notation. -B<x-A<B and solving them separately for x

Step 1 of 3

Given absolute value equation is |3 x-2|<7.

It can be written as -7<3 x-2<7.

To solve for the equality, 3x-2=7 and

$3 x-2=-7$

First, solve the equation 3x-2=7, then add 2 on both sides.

$\begin{aligned}&3 x=7+2 \\&3 x=9\end{aligned}$

Step 2 of 3

Simplify 3x=9 further, by dividing each side with 3 .

$\begin{aligned}&\frac{3 x}{3}=\frac{9}{3} \\&x=3\end{aligned}$

Step 3 of 3

Similarly, 3x-2=-7

From the above term 3x-2=-7,

Add 2 on each side.

$\begin{aligned}&3 x=-7+2 \\&3 x=-5\end{aligned}$

Simplify $3 x=-5$ further, by dividing each side with 3 .

$\begin{aligned}&\frac{3 x}{3}=-\frac{5}{3} \\&x=-\frac{5}{3}\end{aligned}$

Therefore, the solution is $-\frac{5}{3} < x < 3$ or $\left\{-\frac{5}{3}, 3\right\}$

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A 5.8-ft-tall person walks away from a 9-ft lamppost at a constant rate of 3.4 ft/sec. What is the rate that the tip of the pers

Dovator [93]

Answer:

9.56 ft/sec

Step-by-step explanation:

We are told that a 5.8-ft-tall person walks away from a 9-ft lamppost at a constant rate of 3.4 ft/sec.

I've attached an image showing triangle that depicts this;

Thus; dx/dt = 3.4 ft/sec

From the attached image and using principle of similar triangles, we can say that;

9/y = 5.8/(y - x)

9(y - x) = 5.8y

9y - 9x = 5.8y

9y - 5.8y = 9x

3.2y = 9x

y = 9x/3.2

dy/dx = 9/3.2

Now, to find how fast the tip of the shadow is moving away from the lamp post, it is;

dy/dt = dy/dx × dx/dt

dy/dt = (9/3.2) × 3.4

dy/dt = 9.5625 ft/s ≈ 9.56 ft/sec

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3 years ago

What is the equation of the line that passes through (–2, 3) and is parallel to 2x + 3y = 6?

sergeinik [125]

If they are parallel then the coefficients of x and y will remain the same. Only the constant will change.

2x + 3y = ?
2(-2) +3(3) = -4 + 9 = 5 so ? is 5

ANSWER: 2x + 3y = 5

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3 years ago

What is an expression equivalent to m^238/m^-79 in the form m^v, where v is an integer?

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Answer:

$m^{317}$

Step-by-step explanation:

In indices, the division postulate is true:

$\frac{a^m}{a^n} = a^{m-n}$

Apply this to the question:

$\frac{m^{238}}{m^{-79}} \\= m^{238-(-79)}\\= m ^{238+79}\\= m^{317}$

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4 years ago

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3 years ago

Can anyone explain those marked steps in case (iii)

Tema [17]

$\cos\dfrac{3x}2=\cos\left(\dfrac\pi2+\dfrac x2\right)$
$\implies \dfrac{3x}2=2n\pi+\dfrac\pi2+\dfrac x2$

follows from the fact that the cosine function is $2\pi$ -periodic, which means $\cos x=\cos(2\pi+x)$ . Roughly speaking, this is the same as saying that a point on a circle is the same as the point you get by completing a full revolution around the circle (i.e. add $2\pi$ to the original point's angle with respect to the horizontal axis).

If you make another complete revolution (so we're effectively adding $4\pi$ ) we get the same result: $\cos x=\cos(4\pi+x)$ . This is true for any number of complete revolutions, so that this pattern holds for any even multiple of $\pi$ added to the argument. Therefore $\cos x=\cos(2n\pi+x)$ for any integer $n$ .

Next, because $\cos(-x)=\cos x$ , it follows that $\cos x=\cos(2n\pi-x)$ is also true for any integer $n$ . So we have

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The rest follows from considering either case and solving for $x$ .

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3 years ago