Questions: Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. (2x+5)/(5x-2) ≤ 1 Move all terms to the left side of the inequality and then define the left side of the inequality as f(x). Then list the intervals determined by the real zeros of f and the real numbers for which f is undefined. Complete the following table. (Type your answers in interval notation. Use ascending order. Type an integer or a simplified fraction.)

Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. (2x+5)/(5x-2) ≤ 1 Move all terms to the left side of the inequality and then define the left side of the inequality as f(x). Then list the intervals determined by the real zeros of f and the real numbers for which f is undefined. Complete the following table. (Type your answers in interval notation. Use ascending order. Type an integer or a simplified fraction.)
Transcript text: Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. $\frac{2 x+5}{5 x-2} \leq 1$ Move all terms to the left side of the inequality and then define the left side of the inequality as $f(x)$. Then list the intervals determined by the real zeros of $f$ and the real numbers for which $f$ is undefined. Complete the following table. (Type your answers in interval notation. Use ascending order. Type an integer or a simplified fraction.)
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Solution

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Solution Steps

Step 1: Move all terms to the left side of the inequality

The given inequality is

\[ \frac{2x + 5}{5x - 2} \leq 1 \]

Subtract 1 from both sides to move all terms to the left side:

\[ \frac{2x + 5}{5x - 2} - 1 \leq 0 \]

Step 2: Define the left side of the inequality as \( f(x) \)

Define \( f(x) \) as:

\[ f(x) = \frac{2x + 5}{5x - 2} - 1 \]

Simplify \( f(x) \):

\[ f(x) = \frac{2x + 5 - (5x - 2)}{5x - 2} = \frac{2x + 5 - 5x + 2}{5x - 2} = \frac{-3x + 7}{5x - 2} \]

Step 3: Find the real zeros and undefined points of \( f(x) \)

Set the numerator equal to zero to find the real zeros:

\[ -3x + 7 = 0 \implies x = \frac{7}{3} \]

Set the denominator equal to zero to find where \( f(x) \) is undefined:

\[ 5x - 2 = 0 \implies x = \frac{2}{5} \]

Final Answer

The solution set in interval notation is:

\[ \left(-\infty, \frac{2}{5}\right) \cup \left(\frac{7}{3}, \infty\right) \]

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