To determine which inequalities are solved based on the movement from 5 to -10 on a number line, we need to analyze each inequality you provided. We'll solve each one for \( x \) and determine if the solution includes the range from 5 down to -10.
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Inequality: \( \frac{2}{4}x - \frac{4}{5} \geq 1 \frac{1}{5} \)
Simplifying the inequality: \[ \frac{1}{2}x - \frac{4}{5} \geq \frac{6}{5} \]
Adding \(\frac{4}{5}\) to both sides: \[ \frac{1}{2}x \geq \frac{6}{5} + \frac{4}{5} \] \[ \frac{1}{2}x \geq \frac{10}{5} \] \[ \frac{1}{2}x \geq 2 \]
Multiplying both sides by 2: \[ x \geq 4 \] Solution Range: \( x \geq 4 \)
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Inequality: \( \frac{3}{5}x + \frac{2}{5} \geq 3 \frac{2}{5} \)
Simplifying the inequality: \[ \frac{3}{5}x + \frac{2}{5} \geq \frac{17}{5} \]
Subtracting \(\frac{2}{5}\) from both sides: \[ \frac{3}{5}x \geq \frac{17}{5} - \frac{2}{5} \] \[ \frac{3}{5}x \geq \frac{15}{5} \] \[ \frac{3}{5}x \geq 3 \]
Multiplying both sides by \(\frac{5}{3}\): \[ x \geq 5 \] Solution Range: \( x \geq 5 \)
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Inequality: \( \frac{3}{7}x + \frac{1}{7} \leq 1 \frac{6}{7} \)
Simplifying the inequality: \[ \frac{3}{7}x + \frac{1}{7} \leq \frac{13}{7} \]
Subtracting \(\frac{1}{7}\) from both sides: \[ \frac{3}{7}x \leq \frac{13}{7} - \frac{1}{7} \] \[ \frac{3}{7}x \leq \frac{12}{7} \]
Multiplying both sides by \(\frac{7}{3}\): \[ x \leq 4 \] Solution Range: \( x \leq 4 \)
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Inequality: \( \frac{1}{5}x - \frac{3}{5} \leq \frac{2}{5} \)
Simplifying the inequality: \[ \frac{1}{5}x \leq \frac{2}{5} + \frac{3}{5} \] \[ \frac{1}{5}x \leq 1 \]
Multiplying both sides by 5: \[ x \leq 5 \] Solution Range: \( x \leq 5 \)
Summary of Results:
- Inequality 1: \( x \geq 4 \)
- Inequality 2: \( x \geq 5 \)
- Inequality 3: \( x \leq 4 \) (Fits the move from 5 to -10)
- Inequality 4: \( x \leq 5 \)
Given the original movement from 5 to -10, the inequality satisfied during that movement would be Inequality 3, which has the solution range \( x \leq 4 \). This means, indeed, you could have solved that inequality during the transition.