Absolute Values: Functions, Properties

Ans. Finding the absolute values of the given numbers first.

-14, 12, 7, 91, 5, 8, 65, 6

Place the numbers in ascending order now (smallest to the largest number)

-14, 5, 6, 7, 8, 12, 65, 91

Ans. Make f(x)=0. and find the value of x:

f(x)=0

|4x+B|−8=0

|4x+B|=8

Rewrite the following equation as a compound equation and solve each portion separately:

4x + B=8 or 4x + B= 8

4x+B= −8

4x+B−B= −B−8

4x= −B−8

4x÷4= (−B−8)÷4

x= −B−84/4

4x+B= 8

4x+B−B=−B+8

4x=−B+8

4x÷4=(−B+8)÷4

x= −B+84/4

Ans. This can be rewritten as an equation with N as the variable:

N=|N|10

Because a nonnegative number equals its own absolute value, this number must be negative.

|N|=N, and we may rewrite the equation as

N= -N-10

N+N=-N-10+N

2N=-10

2N/2=-10/2

N=-5

The only number that meets the condition is -5.

Ans. |1/n²|

We know the following since -1 < n < 0:

0 < n²  < 1;

-1 < n/2 < 0;

1/n² > 1;

1/n < -1;

0 < n+1 < 1.

Furthermore, we must compare absolute numbers, therefore the greatest must be either |1/n²| or |1/n|.

We also know that when -1 < n < 0, |n²| < |n|.

As a result, we can be certain that |1/n²| >|1/n|.

Ans. Radicals are real numbers that can be represented as powers with rational exponents.

X\(\sqrt{n}\)= n1/x.

If the radical is an even-root radical,

\(\sqrt{x}\) or 4 \(\sqrt{x}\) then the value under the radical has to be positive i.e. x > 0

Hence \(\sqrt{x^2}\) = |x|, because an even-root radical is a positive quantity, as the absolute value. Hence, we use absolute values while solving even-root radicals.

The absolute value symbol is not needed if the radical is an odd-root radical as the value under an odd root radical can be positive or negative.

3\(\sqrt{-8}\) = -2 and 3\(\sqrt{8}\) = 2

Ans. The fundamental rule of absolute value is that any number's absolute value is always positive. If the number is a positive integer, it has a positive absolute value, such as |15| = 15. Even though the number is a negative integer, the absolute value is positive, as shown by |-20| = 20.'

Ans. |x-2|=4 is the given equation. When we eliminate the absolute value sign on one side of the equation, we have the definition of the absolute value function. On the other hand, we obtain a + symbol. + 4 x-2= This yields two equations, each of which we must solve separately.

So, the following equation's answers are x = 6, x = -2.

Ans. We know absolute value always results in non-negative quantities. As a result, we have the following options.

(I) | -13/15| = 13/15 

(II). - |-3| = -(3) = -3 

(III). |2(-3) + 4| = |-6+4| = |-2| = 2

As a result, (I) | -13/15| = 13/15, (II) | -3 | = -3, and (III) |2(-3) + 4| = 2

Ans. Calculating critical numbers.

f'(x) = \(\frac{x^2+x+1-x(2x+1)}{(x^2+x+1)^2}\)

-  \(\frac{x^2+1}{(x^2+x+1)^2}\)

Therefore,

f'(x) = – \(\frac{x^2+1}{(x^2+x+1)^2}\)

Making f’(x) = 0, – x² + 1 = 0 ⇒ x = – 1 or x = 1

So the critical numbers are x = -1 and x = 1

x = 1 ∈ [-2, 0] discarded for the study of absolute extremes.

Now Calculating the absolute extremes, 

At the crucial value x=-1 and the interval's extremes, we evaluate the function.

The absolute maximum is the highest value, and the absolute minimum is the lowest.

We have

f(-2) = –\(\frac{2}{3} \approx\) - 0.67

f(-1) = -1

f(0) = 0.

As a result, the absolute maximum is 0 and occurs at x = 0, while the absolute minimum is -1 and occurs at x=-1.

Ans. 2 | - 2 x - 2| = 16, 

| - 2 x - 2| = 8 | 

The expression - 2 x - 2 equals 8 or - 8 is identical to stating the preceding equation.

1. a) -2 x - 2 = 8

x = - 5 

1. b) - 2 x - 2 = - 8

x = 3 

Check that x = - 5 and x = 3 are solutions to the above problem by substituting them.

Set of solutions: {-5,3}