Solve each rational inequality. Give the solution set in interval notation. 5/(1-x)≤2/(1-x)

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Start by writing down the inequality: \(\frac{5}{1 - x} \leq \frac{2}{1 - x}\).

Identify the domain restrictions by setting the denominator not equal to zero: \(1 - x \neq 0\), which means \(x \neq 1\).

Since the denominators are the same and not zero, subtract \(\frac{2}{1 - x}\) from both sides to get a single rational expression: \(\frac{5}{1 - x} - \frac{2}{1 - x} \leq 0\).

Combine the fractions: \(\frac{5 - 2}{1 - x} \leq 0\), which simplifies to \(\frac{3}{1 - x} \leq 0\).

Analyze the inequality \(\frac{3}{1 - x} \leq 0\) by considering the sign of the denominator \(1 - x\). Since the numerator 3 is positive, the fraction is less than or equal to zero when \(1 - x\) is negative or zero (but zero is excluded due to domain). Solve \(1 - x < 0\) to find the solution set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values that satisfy the inequality while ensuring the denominator is not zero, as division by zero is undefined.

Domain Restrictions

When solving rational inequalities, it is crucial to identify values that make denominators zero and exclude them from the solution set. These restrictions define the domain and prevent invalid solutions.

Interval Notation and Sign Analysis

After simplifying the inequality, use sign analysis to determine where the expression is positive, negative, or zero. Express the solution set using interval notation, which concisely represents all valid values of the variable.

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