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

2/5x−4/5≥ 1 1/5
3/7x+1/7≤ 1 6/7
1/5x−3/5≤ 2/5
3/5x+2/5≥ 3 2/5

a closed circle on the number 5 and an arrow pointing to the left

3 answers

To determine which inequality Hessa solved, we need to analyze the possible solutions and their graphs based on the information provided: she has a closed circle on the number 5 with an arrow pointing to the left. This indicates that the solution set includes 5 and all numbers less than 5.

Let's solve each inequality to check:

  1. \( \frac{2}{5}x - \frac{4}{5} \geq 1 \frac{1}{5} \): \[ \frac{2}{5}x - \frac{4}{5} \geq \frac{6}{5} \] Adding \( \frac{4}{5} \) to both sides: \[ \frac{2}{5}x \geq \frac{6}{5} + \frac{4}{5} \implies \frac{2}{5}x \geq \frac{10}{5} = 2 \] Multiplying by \( \frac{5}{2} \): \[ x \geq 5 \] (This results in an open circle on 5 and an arrow pointing to the right.)

  2. \( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \): \[ \frac{3}{7}x + \frac{1}{7} \leq \frac{13}{7} \] Subtracting \( \frac{1}{7} \): \[ \frac{3}{7}x \leq \frac{12}{7} \] Multiplying by \( \frac{7}{3} \): \[ x \leq 4 \] (This results in a closed circle on 4 with an arrow pointing to the left, not a match.)

  3. \( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \): \[ \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \] Adding \( \frac{3}{5} \): \[ \frac{1}{5}x \leq 1 \] Multiplying by \( 5 \): \[ x \leq 5 \] (This results in a closed circle on 5 with an arrow pointing to the left, which is a match.)

  4. \( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \): \[ \frac{3}{5}x + \frac{2}{5} \geq \frac{17}{5} \] Subtracting \( \frac{2}{5} \): \[ \frac{3}{5}x \geq 3 \] Multiplying by \( \frac{5}{3} \): \[ x \geq 5 \] (This results in an open circle on 5 and an arrow pointing to the right.)

Based on the analysis, the correct inequality that Hessa solved is:

\( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \).

Solve the inequality 9.5x−1.25<32 . Which of the following graphs shows the solution?

To solve the inequality \( 9.5x - 1.25 < 32 \), follow these steps:

  1. Add \( 1.25 \) to both sides: \[ 9.5x < 32 + 1.25 \] \[ 9.5x < 33.25 \]

  2. Now, divide both sides by \( 9.5 \): \[ x < \frac{33.25}{9.5} \]

Calculating \( \frac{33.25}{9.5} \): \[ \frac{33.25}{9.5} = 3.5 \]

So, the solution to the inequality is: \[ x < 3.5 \]

Graphing the Solution

  1. Since the inequality is \( x < 3.5 \), we will have an open circle at 3.5 on the number line (indicating that 3.5 is not included in the solution).
  2. The arrow will point to the left, indicating all values less than 3.5.

Conclusion

Look for a graph with:

  • An open circle at 3.5
  • An arrow pointing to the left.

This graph represents the solution to the inequality \( 9.5x - 1.25 < 32 \).