To solve the first inequality \(-3x + 1 \leq -47\), we can follow these steps:
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Subtract 1 from both sides: \[ -3x \leq -47 - 1 \] \[ -3x \leq -48 \]
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Divide both sides by -3. Remember, when we divide by a negative number, we flip the inequality sign: \[ x \geq \frac{-48}{-3} \] \[ x \geq 16 \]
So the solutions for the inequality \( -3x + 1 \leq -47 \) is \( x \geq 16 \).
To graph this solution on a number line:
- Draw a number line.
- Put a closed circle (or dot) at 16 to signify that 16 is included in the solution (since it's \( \geq\)).
- Shade the line to the right of 16 to show all numbers greater than 16 are also solutions.
Regarding the second part of your question about the student's mistake regarding the inequality \( -4 < x \):
The correct interpretation of \( -4 < x \) is that \( x\) is greater than \(-4\). The correct graph for this inequality should have an open circle at -4 (indicating that -4 is not included) and should extend to the right.
If the student's graph shows \( x < -4\) instead, the mistake likely comes from misunderstanding the direction of the inequality. Therefore, the correct response would be:
The student did \(x\) is less than \(-4\), when the variable is on the other side; \(-4\) is less than \(x\) so \(x\) is greater than (-4.
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