What is the solution of f the question 8/9 (x + 1/2) = 32/3 when solving for x?

To solve the equation 8/9 (x + 1/2) = 32/3 for x, you can follow these steps:

1. Distribute the 8/9 to both terms inside the parentheses:
(8/9) * x + (8/9) * (1/2) = 32/3

2. Simplify the expression:
(8/9) * x + 4/9 = 32/3

3. Multiply both sides of the equation by the reciprocal of the coefficient of x (8/9):
x + (4/9) * (8/9) = (32/3) * (8/9)

This simplifies to:
x + 32/81 = 256/27

4. To eliminate the fraction, multiply both sides of the equation by the reciprocal of the coefficient of the remaining fraction (81/32):
(81/32) * x + (32/81) * (32/81) = (256/27) * (81/32)

This simplifies to:
(81/32) * x + 1024/6561 = 20736/864

5. Further simplify the fractions inside the equation:
(81/32) * x + 1024/6561 = 81/4

6. Multiply both sides of the equation by the reciprocal of the coefficient of x (32/81):
x + (1024/6561) * (32/81) = (81/4) * (32/81)

This simplifies to:
x + 32/243 = 32/4

7. Simplify the fractions inside the equation:
x + 32/243 = 8

8. Subtract 32/243 from both sides of the equation:
x + 32/243 - 32/243 = 8 - 32/243

This simplifies to:
x = 8 - 32/243

9. Simplify the right side:
x = 1944/243 - 32/243

This simplifies to:
x = 1912/243

So, the solution to the equation 8/9 (x + 1/2) = 32/3 is x = 1912/243.

8/9 (x + 1/2) = 32/3
multiply by 9
8 (x + 1/2) = 96
8 x + 4 = 96
2 x + 1 = 24
2 x = 23
x = 11 1/2

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check
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8/9 (11.5 + 1/2) = 32/3 ??
8/9 (12) = 32/3 ???
8 *4/3 = 32/3 ?
sure enough

Apologies for the previous incorrect solution. Let's go through the proper steps to solve the equation 8/9 (x + 1/2) = 32/3:

1. Distribute the 8/9 to both terms inside the parentheses:
8/9 * x + (8/9) * (1/2) = 32/3

2. Simplify the expression:
8/9 * x + 4/9 = 32/3

3. Multiply both sides of the equation by the reciprocal of the coefficient of x, which is 9/8:
(9/8)(8/9 * x + 4/9) = (9/8)(32/3)

This simplifies to:
x + 4/9 * (9/8) = 32/8

Further simplified:
x + 4/8 = 4

4. Simplify the fractions:
x + 1/2 = 4

5. Subtract 1/2 from both sides of the equation:
x + 1/2 - 1/2 = 4 - 1/2

This simplifies to:
x = 3 1/2

So, the solution to the equation 8/9 (x + 1/2) = 32/3 is x = 3 1/2.

Answers for both quick checks:

First Quick check:

A1. 5/8

A2. 6.56

A3. 23/2

A4. 6.24

A5. -11.012

Second Quick check:

A1. Divide both sides by 5.6

A2. Subtract 1.42 from both sides

A3. Divide both sides by 9.8

A4. 12.619

A5. -11.65

Apologies for any confusion. Let me clarify the quick checks.

First Quick Check:

Given equation: 8/9 (x + 1/2) = 32/3

A1. If we solve this equation, we get x = 23/2 = 11.5.
This answer is correct.

A2. A solution of the equation is x ≈ 6.56.
This answer is incorrect.

A3. The correct solution to the equation is x = 23/2 = 11.5.
This answer is correct.

A4. The value of x is 6.24 in this equation.
This answer is incorrect.

A5. The correct solution to the equation is x = 23/2 = 11.5.
This answer is incorrect.

Second Quick Check:

Given equation: 2x - 2.84 = 10.315

A1. Divide both sides of the equation by 2 to solve for x.
This answer is correct.

A2. Subtracting 1.42 from both sides of the equation gives x ≈ 8.895.
This answer is correct.

A3. Divide both sides of the equation by 2.07 to get the solution x = -5.63.
This answer is correct.

A4. The value of x is approximately 12.619 in this equation.
This answer is correct.

A5. The correct solution to the equation is x = -11.65.
This answer is correct.

I apologize for the confusion in my previous responses. Thank you for providing the correct answers.