Use the image to answer the question.

Willow solved an inequality and graphed the solution on the number line. Which of the following inequalities did she solve?

(1 point)
Responses

x - 213
< 216
x - 2 and 1 third < 2 and 1 sixth

x + 1.75 ≤ 5.75
x + 1.75 ≤ 5.75

x + 212
< 212
x + 2 and 1 half < 2 and 1 half

x - 1.5 ≤ 3
x - 1.5 ≤ 3

3 answers

To determine which inequality Willow solved based on the provided options, we need to analyze each inequality and see if it could lead to a graphing scenario on a number line.

  1. x - 213 < 216
    This simplifies to:
    \( x < 216 + 213 \)
    \( x < 429 \)

  2. x - 2 and 1 third < 2 and 1 sixth
    This can be rewritten with fractions:
    \( x - \frac{7}{3} < \frac{13}{6} \)
    Simplifying this gives:
    \( x < \frac{13}{6} + \frac{7}{3} \)
    To combine these fractions, we must find a common denominator. The common denominator of 6 and 3 is 6:
    \( x < \frac{13}{6} + \frac{14}{6} \)
    \( x < \frac{27}{6} \), or \( x < 4.5 \).

  3. x + 1.75 ≤ 5.75
    This simplifies to:
    \( x ≤ 5.75 - 1.75 \)
    \( x ≤ 4 \), which would be a closed dot on 4 extending leftwards.

  4. x + 212 < 212
    This simplifies to:
    \( x < 212 - 212 \)
    \( x < 0 \), which would be an open dot on 0 extending leftwards.

  5. x - 1.5 ≤ 3
    This simplifies to:
    \( x ≤ 3 + 1.5 \)
    \( x ≤ 4.5 \), which would also indicate a closed dot at 4.5 extending leftwards.

To match these inequalities with a possible graph on a number line, we look specifically for the description of points and shading that would occur based on whether it is an open or closed inequality.

Without access to the specific image mentioned, we cannot definitively determine the corresponding graph. However, based on the inequalities, if I had to make an educated guess based solely on general patterns:

  • If the graph shows an interval ending at 4 with a solid dot (closed) at 4, then the option x + 1.75 ≤ 5.75 or x - 1.5 ≤ 3 could be what Willow solved.
  • If the graph shows an open interval ending left of 0 (or later as in the case of other inequalities), then either of those inequalities involving < (less than) would be the resolution.

If you have the graph and it shows, for example, a closed dot at 4 and shading to the left, the answer would be x + 1.75 ≤ 5.75 or x - 1.5 ≤ 3, depending on where it cuts off.

For a clear answer, I would encourage checking the visual of the graph with the options given based on the descriptions.

so whats the answer?

Based on the descriptions of the inequalities provided:

  • If the graph shows a closed dot at 4 with shading to the left, the inequality would be x + 1.75 ≤ 5.75 (which simplifies to \(x ≤ 4\)).

  • If the graph shows a closed dot at 4.5 with shading to the left, then the inequality would be x - 1.5 ≤ 3 (which simplifies to \(x ≤ 4.5\)).

Without seeing the actual graph, the most likely candidate based on closed intervals leading to 4 is x + 1.75 ≤ 5.75.

Therefore, if I had to pick one based on typical scenarios, the most appropriate answer would likely be:

x + 1.75 ≤ 5.75.

If the graph indeed does have a closed circle at 4, that would be the one. If it has a closed circle at 4.5, then you would pick x - 1.5 ≤ 3. But given the first option typically aligns with many scenarios, that would be my final recommendation.