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Twenty Algebra Questions

Please see the attached file.

(Questions 1, 2) Solve the absolute value inequality:

|4x - 4| >= 1
|r - 4.6| < 9

(Question 3) Multiply and simplify:

6y x 4y + 2
12y + 6 5

(Question 4) Find the domain:

f(x) = (x2 + 8x)/(x - 8)

(Question 5) Find the LCM:

63x, 9x2, yx3

(Question 6) Simplify:

y2 + 12y + 35
y2 + 13y + 40

(Questions 7 - 11) Perform the operation and simplify:

7 - 7
x + 7 x - 7

20x3 - 3x2 + 16x + 15 divided by 5x + 3

6 + 1/3
3 - 1/9

1/a + 1
1/a - 1

a2 - 49 divided by a2 - 7a - 8
a2 + 8a + 7 8a2 - 5a - 6

(Questions 12 - 15) Solve:

2 + 8 = 1
x 9

5 = 2 - 1
x x

6 - 8 = 12
y + 5 y - 5 y2 - 25

x = -2x + 2x - 3
2x + 2 4x + 4 x + 1

(Question 16) For the given rational function f, find all values of x which f(x) has the indicated value:

f(x) = (x - 6)/(x - 2); f(x) = 3/2

(Question 17) Solve the formula for the specified letter:

P = A/(a + rt) for r

(Questions 18 - 20) Word problems:

A boat goes 280 miles downstream in the same time it can go 210 miles upstream. The speed of the current is 9 miles per hour. Find the speed of the boat in still water.

The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 5 ft from a light is 3 foot candles, find the intensity on a screen 15 ft from the light.

One maid can clean the house 3 times faster than another. Working together they can clean the entire house in 3 hours. How long would it take the faster maid cleaning alone?

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Solution Preview

***In the second problem, the solution says, "You would graph this on a number line by putting solid dots at -4.4 and 13.6, then shading between them." This should read "open dots".***

Solve the absolute value inequality

In this problem, the values of 4x - 4 are greater than or equal to 1 (so 4x - 4 could equal 1, 2, 3, etc.) or less than or equal to -1 (so 4x - 4 could equal the opposite of the previous examples: -1, -2, -3, etc.). Mathematically, this looks like:

4x - 4 &#8805; 1 OR 4x - 4 &#8804; -1
4x &#8805; 5 4x &#8804; 3
x &#8805; 5/4 x &#8804; ¾

The answer is: x &#8804; ¾ or x &#8805; 5/4.

You can graph this on a number line by putting a solid dot at ¾ (0.75) and shading to the left and putting a solid dot at 5/4 (1.25) and shading to the right.

Here, the values of r - 4.6 are less than or equal to 9 and greater than or equal to -9.

r - 4.6 &#8805; -9 AND r - 4.6 &#8804; 9
r &#8805; -9 + 4.6 r &#8804; 9 + 4.6
r &#8805; -4.4 r &#8804; 13.6

The answer is: r &#8805; -4.4 and r &#8804; 13.6

You would graph this on a number line by putting solid dots at -4.4 and 13.6, then shading between them. Multiply and simplify

When multiplying fractions, you simply multiply the numerators together and multiply the denominators together. (Multiplying fractions is easier than adding fractions. When adding fractions, the denominators need to be the same. See the problems below marked "Solve" for examples.)

6y x 4y + 2
12y + 6 5

(6y)(4y + 2)
(12y + 6)(5)

24y2 + 12y
60y + 30

To simplify, you can factor a 12y from the numerator and a 30 from the denominator:

12y(2y + 1)
30(2y + 1)

Now, you can see that there is a 2y + 1 on the top and the bottom, so you can cancel those out. Similarly, you can cancel out a 6 (6 goes into both 12 and 30):

12y(2y + 1)
30(2y + 1)

12y
30

2y/5

The answer is 2y/5

Find the domain

The domain consists of the values of x that can be used in the function. You can't divide by 0, so the domain is all values of x that don't make the denominator equal to 0. So, we have to find the values of x for which the denominator does equal 0; then the domain will be all possible x's (all real number) except for those.

x - 8 = 0
x = 8

The domain of f(x) is all real numbers except for x = 8. Find the LCM

LCM stands for least ...

Solution Summary

This problem set consists of 20 algebra questions involving:

-- absolute value inequalities
-- simplifying rational expressions
-- the domain of a function
-- the least common multiple of terms
-- solving rational equations
-- word problems

The solution includes answers to all of the questions as well as detailed, step-by-step explanations.

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