Question

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

\[
\boxed{3}
\]