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Question

A number line is shown ranging from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line between 2 and 3, just after the 2. A leftward arrow is marked on the number line that originates at the closed point and goes beyond negative 10.

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

(1 point)
Responses

3.7x−1.4≥6.37
3.7 x minus 1.4 greater than or equal to 6.37

3.2x−1.4≤5.32
3.2 x minus 1.4 less than or equal to 5.32

4.9x+1.2≥11.49
4.9 x plus 1.2 greater than or equal to 11.49

2.8x+0.9≤6.5

Answer 1

To determine which inequality corresponds to Amari's solution on the number line, we first need to analyze the information provided. The closed point plotted between 2 and 3 indicates that the solution includes this value, meaning the inequality is likely to be inclusive.

Since the closed point is between 2 and 3, we need to find which inequality would yield a solution set that includes numbers around this range (considering the leftward arrow extending into the negative numbers).

Now, let's evaluate the inequalities one by one:

1. 3.7x - 1.4 ≥ 6.37

   • Solving this inequality:
     • Add 1.4 to both sides: 3.7x ≥ 7.77
     • Divide by 3.7: x ≥ 2.1
   • This indicates that the solution set starts from 2.1 and goes to positive infinity. This does not match the leftward arrow.

2. 3.2x - 1.4 ≤ 5.32

   • Solving this inequality:
     • Add 1.4 to both sides: 3.2x ≤ 6.72
     • Divide by 3.2: x ≤ 2.1
   • This indicates that the solution set includes values less than or equal to 2.1, which matches the leftward arrow extending from the region between 2 and 3.

3. 4.9x + 1.2 ≥ 11.49

   • Solving this inequality:
     • Subtract 1.2 from both sides: 4.9x ≥ 10.29
     • Divide by 4.9: x ≥ 2.1
   • Similar to the first, this does not match the left quadrant or indicate values extending leftward.

4. 2.8x + 0.9 ≤ 6.5

   • Solving this inequality:
     • Subtract 0.9 from both sides: 2.8x ≤ 5.6
     • Divide by 2.8: x ≤ 2
   • This indicates that the solution includes values less than or equal to 2, which is also reasonable for leftward movement, but not enough to match the closed point between 2 and 3.

Based on this analysis, the inequality that Amari would have solved, which corresponds to a closed dot between 2 and 3 and a leftward arrow, is:

3.2x - 1.4 ≤ 5.32