Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

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Identify the critical points by setting the numerator and denominator equal to zero separately: solve \( x + 3 = 0 \) and \( x + 4 = 0 \). These points divide the number line into intervals.

Determine the intervals created by the critical points. In this case, the critical points are \( x = -3 \) and \( x = -4 \), so the intervals are \( (-\infty, -4) \), \( (-4, -3) \), and \( (-3, \infty) \).

Test a sample value from each interval in the inequality \( \frac{x+3}{x+4} < 0 \) to check whether the expression is negative in that interval.

Based on the sign of the expression in each interval, select the intervals where the inequality holds true (where the expression is less than zero).

Express the solution set in interval notation, excluding any points where the denominator is zero (since the expression is undefined there), and then graph the solution on a real number line.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Inequalities

Rational inequalities involve expressions where one rational expression is compared to zero or another expression using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine whether the rational expression is positive or negative, which helps identify where the inequality holds.

Interval Notation and Graphing Solutions

After determining the solution intervals, express them using interval notation to clearly show the range of values satisfying the inequality. Graphing on a number line visually represents these intervals, indicating included or excluded points based on inequality strictness.