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ololo11 [35]
3 years ago
5

|5x+8|=|10x+7| solve the equation.

Mathematics
1 answer:
liberstina [14]3 years ago
7 0

We have to take care of absolute values. The absolute value of an expression is the positive version of that expression, i.e.

|x| = \begin{cases} x &\text{ if } x \geq 0\\ -x &\text{ if } x < 0 \end{cases}

So, first of all, we must observe that 5x+8 is positive if

5x+8 \geq 0 \iff 5x \geq -8 \iff x \geq \dfrac{-8}{5}

and similarly,

10x+7 \geq 0 \iff 10x \geq -7 \iff x \geq \dfrac{-7}{10}

So, since

\dfrac{-8}{5} < \dfrac{-7}{10}, we can divide the number line in three zones:

Zone 1: x < -8/5

In this zone, both expressions are negative. This means that

|5x+8|= -5x-8,\qquad |10x+7| = -10x-7

In fact, as we already said, the absolute value flips the sign of an expression if that expression is negative. So, the equation becomes

-5x-8 = -10x-7 \iff 5x=1 \iff x = \dfrac{1}{5}

But we can't accept this solution (yet), because we're supposing x < -8/5.

Zone 2: -8/5 < x < -7/10

In this zone, 5x+8 has become positive, while 10x+7 is still negative. This means that

|5x+8|= 5x+8,\qquad |10x+7| = -10x-7

So, the equation becomes

5x+8 = -10x-7 \iff 15x=-15 \iff x = -1

Since indeed -8/5 < -1 < -7/10, we can accept this solution.

Zone 3: x > -7/10

In this zone, both expressions are positive, which means that the absolute value changes nothing:

|5x+8|= 5x+8,\qquad |10x+7| = 10x+7

So, the equation becomes

5x+8 = 10x+7 \iff 5x=1 \iff x = \dfrac{1}{5}

Since indeed 1/5 > -7/10, we can accept this solution.

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A family buys a studio apartment for $180.00 They pay a down payment of $54.00, Their down payment is what percent of the purcha
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Answer:

Part A: 30%

Part B: 45000%

Step-by-step explanation:

To get your answer, you have to divide whatever number the down payment is by the number the family is supposed to pay for the studio apartment.

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A lottery game has balls numbered 1 through 15. What is the probability of selecting an even numbered ball or a 13?
LiRa [457]

Answer:

8 / 15

Step-by-step explanation:

Sample space = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

Required outcome :

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Answer: (15.47, 24.53)

Step-by-step explanation:

We know that the confidence interval for population mean is given by :_

\overline{x}\pm z^*\dfrac{\sigma}{\sqrt{n}}

, where n= sample size.

\sigma = standard deviation.

\overline{x}= sample mean.

z*= Critical value.

Given : n= 450

\overline{x}=20

\sigma=49

Critical value for 95% confidence = z*=1.96     [From z-value table]

Then, the 95% confidence interval will be :-

20\pm (1.96)\dfrac{49}{\sqrt{450}}

\approx 20\pm (1.96)(2.31)

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Hence, the 95% confidence interval for the mean change in score μ μ in the population of all high school seniors. : (15.47, 24.53)

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