A few pre-calc questions? please?

Question

1. solve for "x"

1=((x+2)/3)-((x-1)/5)

2. solve for "x"

|3x-5|=6

3. Solve the inequality. Write the solution in interval notation.

(5/3)-(1/4x)>(2/3x)-(1/6)

4. Solve the inequality. Write solution in interval notation.

1/2|x-6|-2 ≥ 2


Thank you so much!!!!!

Answer 1

1 = ((x + 2) / 3) - (x - 1) / 5

Multiply each side by 15

1(15) = 15[((x + 2) / 3) - (x - 1) / 5]

15 = 5(x + 2) - 3(x - 1)

Distributive property : a(b + c) = ab + ac , a(b - c) = ab - ac

15 =5x + 10 -3x +3

15 =2x +13

Subtract 13 to each side

15 - 13 = 2x +13 - 13

2 = 2x + 0

2 = 2x

Divide each side by 2

1 = x

Symetric property : a = b then b =a

x = 1.

 

Answer 2

2.

|3x - 5| = 6

Case 1 :

3x  - 5 = 6

Add 5 to each side

3x -5 +5 = 6 + 5

3x + 0 = 11

3x = 11

Divide each side by 3

3x / 3 = 11 / 3

x = 11 / 3.

Case 2 :

3x - 5 = -6

Add 5 to each side

3x - 5 + 5 = -6 + 5

3x + 0 = -1

3x = -1

Divide each side by 3

3x  / 3 = -1 / 3

There fore x = -1 / 3.

 

Answer 3

3) The inequality is

• Step-1

State the exclude values,These are the values for which denominator is zero.

The exclude value of the inequality is 0.

• Step - 2

Solve the related equation

Solution of related equation x   = 1/2.

• Step - 3

Draw the vertical lines at the exclude value and at the solution to separate the number line into intervals.

 

Answer 4

Continuous...

• Step - 4

Now test  sample values in each interval to determine whether values in the interval satisify the inequality.

Test interval     x - value      Inequality                                          Conclusion

(-∞, 0)                      x =  -1         (5/3)-[1/4(-1)] > [2/3(-1)]-(1/6)⇒1.91>0.832         True

(0, 1/2)                     x = 0.1        (5/3)-[1/4(0.1)] > [2/3(0.1)]-(1/6)⇒ -0.83 > -0.10    False

(1/2, ∞)                   x = 0.8         (5/3)-[1/4(0.8)] > [2/3(0.8)]-(1/6)⇒1.35 > 0.36       True

The above conclude that the inequality is satisfied for all x - values in (-∞, 0) and (1/2, ∞).

This implies that the solution  of  the  inequalityis  the  interval (-∞, 0) and (1/2, ∞) . as shown in Figure below. Note that the original inequality contains a “less than” symbol. This means that the solution set does not contain the endpoints of the test intervals are (-∞, 0) and(1/2, ∞)

The solution of the inequality is x  < 0 or x  > 1/2

The interval notation from of solution is (-∞, 0) U (1/2, ∞).

 

Answer 5

4) The absolute inequality is 1/2|x - 6| - 2 ≥ 2

Add 2 to each side

1/2|x - 6| ≥ 4

Multiply each side by 2.

|x - 6| ≥ 8

|x| ≥ a can be written as x ≥ a or x ≤ - a

|x - 6| ≥ 8 can be written as x - 6 ≥ 8 or x - 6 ≤ - 8

Solve the inequality : x - 6 ≥ 8

Add 6 to each side.

x - 6 + 6 ≥ 8 + 6

x  ≥ 14

Solve the inequality : x - 6 ≤ - 8

Add 6 to each side.

x - 6 + 6 ≤ - 8 + 6

x ≤ - 2

Therefore the solution of the absolute inequality is x  ≥ 14 or x ≤ - 2

Solution set is {x Є R| x ≤ -2 or x ≥ 14}

Solution in interval notation form (-∞, -2] U [14, ∞).

Observe the graph , the closed circle means that -2 and 14 are the solutions of the inequality.