Questions: Solve the nonlinear inequality. Express the solution using interval notation. 15/(x-1) - 16/x ≥ 1 Graph the solution set.

Solution Steps

Subtract 1 from both sides of the inequality to get:

$\frac{15}{x-1} - \frac{16}{x} - 1 \ge 0$

Find a common denominator and combine the fractions:

$\frac{15x - 16(x-1) - x(x-1)}{x(x-1)} \ge 0$

$\frac{15x - 16x + 16 - x^2 + x}{x(x-1)} \ge 0$

$\frac{-x^2 + 16}{x(x-1)} \ge 0$

Multiply both sides by -1 (and reverse the inequality sign) to get:

$\frac{x^2 - 16}{x(x-1)} \le 0$

Factor the numerator:

$\frac{(x-4)(x+4)}{x(x-1)} \le 0$

The critical values are x = -4, 0, 1, and 4. These are the values that make the numerator or denominator equal to zero. We need to test the intervals determined by these values.

• $x < -4$: Choose $x = -5$. The expression becomes $\frac{(-9)(-1)}{(-5)(-6)} = \frac{9}{30} > 0$, so this interval is not part of the solution.
• $-4 < x < 0$: Choose $x = -1$. The expression becomes $\frac{(-5)(3)}{(-1)(-2)} = \frac{-15}{2} < 0$, so this interval is part of the solution.
• $0 < x < 1$: Choose $x = \frac{1}{2}$. The expression becomes $\frac{(\frac{-7}{2})(\frac{9}{2})}{(\frac{1}{2})(\frac{-1}{2})} = \frac{\frac{-63}{4}}{\frac{-1}{4}} = 63 > 0$, so this interval is not part of the solution.
• $1 < x < 4$: Choose $x = 2$. The expression becomes $\frac{(-2)(6)}{(2)(1)} = -6 < 0$, so this interval is part of the solution.
• $x > 4$: Choose $x = 5$. The expression becomes $\frac{(1)(9)}{(5)(4)} = \frac{9}{20} > 0$, so this interval is not part of the solution.

Final Answer: