Read the problem below. The problem was solved incorrectly. Solve the problem correctly. Then explain the error that was made.

Question

Problem
3x + 2x - 6 = 24
-2x-2x
X-6=24
+6
+6
X = 30

Answer 1

Let's solve the equation correctly first:

The original equation is: \[ 3x + 2x - 6 = 24 \]

1. Combine like terms on the left side: \[ (3x + 2x) - 6 = 24 \] \[ 5x - 6 = 24 \]

2. Add 6 to both sides to isolate the term with \(x\): \[ 5x - 6 + 6 = 24 + 6 \] \[ 5x = 30 \]

3. Divide both sides by 5 to solve for \(x\): \[ x = \frac{30}{5} \] \[ x = 6 \]

So, the correct solution is: \[ x = 6 \]

Explanation of the Error:

In the initial incorrect solution, the expression was mismanaged. The steps outlined do not correctly follow the basic properties of equality or algebra. Here are some specific errors:

1. Combining terms incorrectly: The initial step shows an assumption that \( -2x - 2x \) (which is unclear) was a step to reduce the left side; however, the expression actually contained \(3x + 2x - 6\). The terms needed to be combined properly as \(5x - 6\).

2. Misinterpretation of equality: The transformation from \(5x - 6 = 24\) to \(X - 6 = 24\) was not valid. The variable was not isolated correctly.

3. Incorrect operations: The conclusion of \(X = 30\) suggests a misunderstanding of operations when solving for \(x\). The variable should have been simplified and solved from \(5x = 30\) to \(x = 6\), instead of directly jumping to \(X = 30\).

Overall, the mistakes stemmed from improper algebraic manipulation and misunderstanding of solving simple linear equations.

Answer 2

3x + 2x - 6 = 24

Answer 3

Let's solve the equation step-by-step correctly:

Given: \[ 3x + 2x - 6 = 24 \]

Step 1: Combine Like Terms

First, combine \(3x\) and \(2x\):

\[ 5x - 6 = 24 \]

Step 2: Add 6 to Both Sides

Next, add \(6\) to both sides to isolate the term with \(x\):

\[ 5x - 6 + 6 = 24 + 6 \]

\[ 5x = 30 \]

Step 3: Divide by 5

Finally, divide both sides by \(5\) to solve for \(x\):

\[ x = \frac{30}{5} \]

\[ x = 6 \]

Conclusion

Thus, the solution to the equation \(3x + 2x - 6 = 24\) is:

\[ \boxed{6} \]

If you have more questions or need clarifications, feel free to ask!

Answer 4

Demetrius can spend as much as `\$50` on shirts.

Shirts, `s`, cost `\$16` each at a nearby store.

Answer 5

To find out how many shirts Demetrius can buy with a budget of $50, we can set up the equation using the cost of each shirt.

1. Let \( s \) be the number of shirts Demetrius can buy.
2. Each shirt costs $16, so the total cost of \( s \) shirts is \( 16s \).
3. Since he can spend no more than $50, we can write the following inequality:

\[ 16s \leq 50 \]

Step 1: Solve the Inequality

To find out how many shirts he can afford, divide both sides of the inequality by 16:

\[ s \leq \frac{50}{16} \]

Calculating \( \frac{50}{16} \):

\[ \frac{50}{16} = 3.125 \]

Since \( s \) must be a whole number (you can't buy a fraction of a shirt), we take the largest whole number that is less than or equal to 3.125:

\[ s \leq 3 \]

Conclusion

Demetrius can buy a maximum of 3 shirts with his budget of $50.

Verification:

• Cost for 3 shirts: \[ 3 \times 16 = 48 \]

• Cost for 4 shirts: \[ 4 \times 16 = 64 \]

Since $48 is within the budget and $64 exceeds it, the conclusion is confirmed. Thus, the maximum number of shirts he can buy is:

\[ \boxed{3} \]