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Solve the inequality 2/3x≤5

. Which of the following graphs shows the solution? (1 point)
Responses

A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 5 start fraction 2 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.
Image with alt text: A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 5 start fraction 2 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.

A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 4 start fraction 1 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.
Image with alt text: A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 4 start fraction 1 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.

A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 3 start fraction 1 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.
Image with alt text: A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 3 start fraction 1 over 3 end fraction. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.

A number line ranges from negative 10 to 10 with numbers listed in unit increments. A closed point is plotted on the number line at 7.5. A leftward arrow is marked on the number line that originates at the closed circle and goes beyond negative 10.

1 answer

To solve the inequality \(\frac{2}{3}x \leq 5\), we can first isolate \(x\).

  1. Multiply both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\): \[ x \leq 5 \cdot \frac{3}{2} \]

  2. Calculate \(5 \cdot \frac{3}{2}\): \[ 5 \cdot \frac{3}{2} = \frac{15}{2} = 7.5 \]

So the solution to the inequality is: \[ x \leq 7.5 \]

This means the solution set includes all values of \(x\) that are less than or equal to \(7.5\).

Now, looking at the graph options you provided, you need to find the one that shows a closed point at \(7.5\) with a leftward arrow, indicating all numbers to the left of \(7.5\).

The correct graph should have:

  • A closed point at \(7.5\)
  • A leftward arrow extending towards negative infinity.

Based on your description, the fourth option is the correct one: "A number line ranges from negative 10 to 10 with a closed point plotted at 7.5 and a leftward arrow originating at the closed circle and going beyond negative 10."

Therefore, the answer is the fourth graph.