In the given steps, the mistake occurs when the student divides both sides of the inequality by (x + 2/5). Let's go through the steps and explain why it is incorrect:
1. Start with the inequality: 5x + 2 ≥ x + 2/5
2. The student tries to factor out 5 on the left side: 5(x + 2/5) ≥ x + 2/5
3. Now, the mistake happens. Dividing both sides by (x + 2/5) is not valid. When dividing inequalities, we need to consider the sign of the divisor.
To understand why, let's consider a simple example: 2 > 1. If we divide both sides by 1, we get 2/1 > 1/1, which is still true. However, if we divide both sides by 2, we get 2/2 > 1/2, which is not true. Dividing by a variable or an expression that could be zero or change signs can introduce extraneous solutions.
4. Therefore, dividing both sides by (x + 2/5) is incorrect.
To find the correct solution set, we need to follow a different approach:
1. Start with the inequality: 5x + 2 ≥ x + 2/5
2. Subtract x from both sides to isolate the variable term: 5x - x + 2 ≥ x - x + 2/5, which simplifies to 4x + 2 ≥ 2/5.
3. Now, subtract 2 from both sides to isolate the variable term: 4x + 2 - 2 ≥ 2/5 - 2, which simplifies to 4x ≥ 2/5 - 2.
4. Simplify the right side: 2/5 - 2 = -10/5 + 2/5 = -8/5.
5. Divide both sides by 4 to solve for x: (4x)/4 ≥ (-8/5)/4, which simplifies to x ≥ -2/5.
6. So, the correct solution set is x ≥ -2/5.
To summarize, the mistake in the student's steps is dividing both sides of the inequality by (x + 2/5). The correct solution set is x ≥ -2/5.