The distance between two real numbers $x,y$ is the absolute value of their difference, $|x-y|$ . So the statement "all $x$ that are 1/4 away from -6" is captured in the equation

$|x-(-6)|=\dfrac14$

$|x+6|=\dfrac14$

To solve, you can use the definition of absolute value: if $x$ is not negative, then $|x|=x$ ; otherwise, $|x|=-x$ .

So there are two cases to consider:

1. Suppose $x+6\ge0$ . Then $|x+6|=x+6$ , and the equation reduces to

$x+6=\dfrac14\implies x=-\dfrac{23}4$

2. Suppose $x+6$ . Then $|x+6|=-(x+6)=-x-6$ and

$-x-6=\dfrac14\implies x=-\dfrac{25}4$

$-6=-\dfrac{24}4$ , so the two solutions we found are correct because both $-\dfrac{23}4$ and $-\dfrac{25}4$ are indeed 1/4 away from -6.

- - -

For the question regarding inequality, you know that the absolute value must return a non-negative number. In other words, there is no $x$ such that $|x|$ .

6 0

3 years ago

What are the solutions of -x^2+4x=x-4

Stolb23 [73]

Answer:

x = -1 and x = 4

Step-by-step explanation:

Step 1: Get everything to one side of the equation so it's set equal to zero..

-x² + 4x = x - 4 becomes...

-x² + 3x + 4 = 0 (subtract x and add 4 to both sides)

x² - 3x - 4 = 0 (divide both sides by -1, we want the x² term to be positive)

(x - 4)(x + 1) = 0 (factor)

x - 4 = 0, then x = 4

and

x + 1 = 0, then x = -1

8 0

3 years ago

Visualize the following procedure: Tear in half a square piece of paper with an area of one square unit. Then tear it in half ag

Nadusha1986 [10]

Answer:

$Area = \frac{1}{16}$ square units

Step-by-step explanation:

To predict the area of the piece of paper after 5 tears we can use a geometric sequence.

Each time a piece is torn, half of the previous area is lost. If the area of the first piece is 1 square unit then:

$a_1 = 1\\\\r = \frac{1}{2}\\\\a_2 = 1 *\frac{1}{2}\\\\a_2 = \frac{1}{2}\\\\a_n = 1(\frac{1}{2})^{n-1}\\\\a_5 = 1(\frac{1}{2})^{5-1}$

$a_5 = \frac{1}{16}$ square units

4 0

4 years ago

Read 2 more answers

The ratio of boys to girls in a class 4 : 5<br><br> A) What fraction of the class are boys

Travka [436]

Answer:

4/9

Step-by-step explanation:

Ratio of boys = 4

Ratio of girls = 5

1. When illustrating ratios in fraction, you add the ratios

4 + 5 = 9

2. Next you put the boys ratio over the total ratio

4/9

3. Sometimes it can be reduced to a lower form eg. 5/10 can be reduced to

1/2.

3 0

3 years ago

1. The temperature of a body rises from -6°c to 5°c. By how much degrees does it rises? ​

1. The temperature of a body rises from -6°c to 5°c. By how much degrees does it rises?