Question

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

Accepted Answer

Rewrite the inequality in standard form by subtracting 5 from both sides: $$ \frac{x + 3}{x - 4} - 5 \leq 0 . $$Combine the terms into a single fraction by finding a common denominator: $$ \frac{x + 3 - 5(x - 4)}{x - 4} \leq 0 . $$Simplify the numerator of the fraction: $$ x + 3 - 5(x - 4) = x + 3 - 5x + 20 = -4x + 23 . $$The inequality becomes $$ \frac{-4x + 23}{x - 4} \leq 0 . $$Determine the critical points by setting the numerator and denominator equal to zero: $$ -4x + 23 = 0 $$ gives $$ x = \frac{23}{4} $$, and $$ x - 4 = 0 $$ gives $$ x = 4 . $$These critical points divide the number line into intervals. Test the sign of the fraction in each interval created by the critical points (e.g., $$ (-\infty, 4) $$, $$ (4, \frac{23}{4}) $$, and $$ (\frac{23}{4}, \infty) ) by $$substituting test values into the fraction $$ \frac{-4x + 23}{x - 4} . $$Determine where the fraction is less than or equal to zero. Include the critical points in the solution set if they satisfy the inequality. For $$ x = \frac{23}{4} $$, the numerator is zero, so it satisfies the inequality. For $$ x = 4 $$, the denominator is zero, so it is excluded. Represent the solution set on a real number line, shading the appropriate intervals and marking any included points.

In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inequalities

Rational Expressions

Graphing Solution Sets