MATH FINAL!! Intermed. Algebra help!! Inequalities??

Question

Okay so im stuck here..can you EXPLAIN how to do it?? i have the answers but dont know how to get them..it says :solve inequalities and graph using interval notation

1. 3/2 I 2x-5 I -2= 7 (the answer is x= -1/2 and 11/2)
(that is 3/2 absolute value 2x-5 then -2 = 7)

2. -5 < 1/4 (x-4) -2(4-x) (answer is 16/9)

3. 2-x (less than or equal to) 3-(5x-4) (answer is 5/4)

Answer 1

1) 3/2 I 2x-5 I -2= 7

Isolate the absolute value symbols

Add 2

3/2 | 2x - 5 | = 9

Multiply by 2/3

| 2x - 5 | = 6

If the absolute value of a number is 6, then the number must be -6 or 6 (because they are each 6 units from zero).

2x - 5 = 6
2x = 11
x = 11/2

2x - 5 = -6
2x = -1
x = -1/2

2) -5 < 1/4 (x-4) -2(4-x)

Use the distributive property

-5 < x/4 - 1 - 8 + 2x

Multiply by 4 to eliminate the fraction

-20 < x - 4 - 32 + 8x

Combine like terms

-20 < 9x - 36

Add 36

16 < 9x

Divide by 9

16/9 < x

or

x > 16/9

3) 2 - x < 3 - (5x - 4)

Distribute

2 - x < 3 - 5x + 4

Combine like terms

2 - x < 7 - 5x

Add 5x

2 + 4x < 7

Subtract 2

4x < 5

Divide by 4

x < (or equal to) 5/4

Answer 2

An absolute sign is usually a tip-off you have a positive and negative result. For the positive one, drop the absolute sign and solve. You should get the 11/2 result. Now, just consider the 2x-5 term inside the absolute signs. That is equal to 6. To find the negative x, set 2x-5 to -6, and out pops your -1/2.

Change the inequality to an equation and solve.
-5 = 1/4 (x-4) + 2(x-4) [I did a sneaky sign-change], but notice we have two x-4 terms
-5 = 9/4 (x-4) and (x-4) = -20/9
Then x = -20/9 + 36/9 = 16/9

This should give you the hang. Try # 3

Answer 3

1) (3/2) |2x - 5| - 2 = 7

Your goal is to isolate the |2x - 5|, because for an equation |y| = c, it follows that y = c or y = -c.

(3/2) |2x - 5| = 9

Multiply both sides by (2/3),

|2x - 5| = 9(2/3)
|2x - 5| = 6

This splits up into two equations:
2x - 5 = 6
2x - 5 = -6

Solve each individually.
2x - 5 = 6, 2x = 11, x = 11/2.
2x - 5 = -6, 2x = -1, x = -1/2

Therefore, x = {-1/2, 11/2}

2) -5 < (1/4)(x - 4) - 2(4 - x)

Your goal is to isolate x. Expand the right hand side first,

-5 < (1/4)x - 1 - 8 + 2x

-5 < (1/4)x - 9 + (8/4)x
-5 < (9/4)x - 9
4 < (9/4)x
4(4/9) < x
16/9 < x

Which can also be stated as x > 16/9.
Therefore, your solution set is
x > 16/9

3) 2 - x <= 3 - (5x - 4)

First, expand the right hand side by distributing the minus.

2 - x <= 3 - 5x + 4

The goal is to bring all x terms to the left hand side, and everything else to the right hand side.

-x + 5x <= 3 - 2 + 4
4x <= 5
x <= 5/4