Question 3.5.11: Solving a Rational Inequality Solve the inequality. 1 - x/x ...

Graph the related rational function

$y=\frac{1 – x}{x + 4}$

The real solutions of $\frac{1 – x}{x + 4} ≥ 0$ are the x-values for which the graph lies above or on the x-axis. This is true for all x to the right of the vertical asymptote at x = -4, up to and including the x-intercept at (1, 0). Therefore, the solution set of the inequality is (-4, 1].

By inspecting the graph of the related function, we can also determine that the solution set of $\frac{1 – x}{x + 4} < 0$ is (-∞, -4) ∪ (1, ∞) and that the solution set of the equation $\frac{1 – x}{x + 4} = 0$ is {1}, the x-value of the x- intercept. (This graphical method may be used to solve other equations and inequalities including those defined by polynomials.)