Help?!? math! 10 points

Question

Solve the equation or inequality by first clearing the fractions or the Decimals.

1..... 1/3x - 4 < 5/6
2.... -3/8x + 1/4  _>  -7/12
3.... 4.2x + 3.5 = 16.1
4..... 8.5x - 6.97=0

5.... Each letter in the word VACATION is written on a separate slip of paper and placed in a hat. A letter is chosen at random from the hat. What is the probability that the chosen letter is a vowel?

6..... A spring sweater is on sale for 15% off the original price of $35. What is the sale price of the sweater?

Answer 1

1) The inequality is 1/3x - 4 < 5/6.

Add 4 to each side.

1/3x - 4 + 4 < 5/6 + 4

1/3x + 0 < (5+24) / 6             [Since leadt common multiple is 6]

1/3x < 29 / 6

Multiply by 3 each side.

(1/3x )*3  < (29 / 6) *3

Cancel common terms.

x < 29 / 2.

2) The inequality is -3/8x + 1/4  ≥  -7/12.

Subtract 1/4 from each side.

-3/8x + 1/4 - 1/4 ≥  -7/12 - 1/4

-3/8x + 0 ≥ (-7 -3) / 12               [Since Least Common Multiple of 12 and 4 is 12]

-3/8x + 0 ≥ -10 / 12

Muliply each side by -8/3.

If a ≥ b then a * -c ≤ b * -c.

(-3/8x)(-8/3) ≤ (-10/12)(-8/3)

x ≤ (-10*-8) / (12*3)

x ≤ 80 / 36

x ≤ 20 / 9.

3) The equation is 4.2x + 3.5 = 16.1.

Subtract 3.5 from each side.

4.2x + 3.5 - 3.5 = 16.1 - 3.5

4.2x + 0 = 12.6

4.2x = 12.6

Divide each side by 4.2.

4.2x /4.2 = 12.6/4.2

Cancel common terms.

x = 12.6/4.2

x = 3.

4) The equation is 8.5x - 6.97 = 0.

Add 6.97 to each side.

8.5x - 6.97 + 6.97 = 0 + 6.97

8.5x = 6.97

Divide each side by 8.5.

8.5x /8.5 = 6.97/8.5

Cancel common terms.

x = 6.97/8.5

x = 0.82

5) Vowels are a, e, i, o, u.

Number of vowels in the word VACATION is 4.

Total number of letters in the word = 8.

Therefore probability of getting a vowel = 4/8 = 1/2.

6) Original price of spring sweater =  $35.

Discount = 15% of 35 = 15/100 * 35 = 15*35 / 100 = 525/100 = $ 5.25.

Sale price of spring sweater = Original price - Discount =  $35 - $ 5.25 = $29.75.