Rational inequalities
In a rational inequality the unknown appears in a denominator. The temptation is to "get rid of" the fraction by cross-multiplying, but that is risky: you do not know whether what you are multiplying by is positive or negative. The correct method is different: study the sign of the numerator and the denominator separately, using a sign table.
Why we do not cross-multiply
Solve the following rational inequality, expressing its solution as an interval: $\frac{x-4}{x+3} > 0$
Solve the following rational inequality, expressing its solution as an interval:
$\frac{x-4}{x+3} > 0$
Solve the inequality: $\frac{x}{x-5} < 0$
Solve the inequality:
$\frac{x}{x-5} < 0$
The denominator is never equal to zero
Determine the values of $x$ that satisfy: $\frac{4}{x-6} \le 0$
Determine the values of $x$ that satisfy:
$\frac{4}{x-6} \le 0$
Solve the following inequality: $\frac{x^2-9}{x+1} > 0$
Solve the following inequality:
$\frac{x^2-9}{x+1} > 0$
When there is a term on the other side
If the inequality is $\frac{P(x)}{Q(x)}\gtrless k$ (with $k\neq0$) or has fractions on both sides, the first step is to move everything to one side and combine into a single fraction, never cross-multiply.
Find the solution set of the inequality: $\frac{x+3}{x-2} \ge 2$
Find the solution set of the inequality:
$\frac{x+3}{x-2} \ge 2$
Determine the set of values that satisfy the following inequality between algebraic fractions: $\frac{1}{x+1} < \frac{1}{x-3}$
Determine the set of values that satisfy the following inequality between algebraic fractions:
$\frac{1}{x+1} < \frac{1}{x-3}$
Problems: average cost, speed, concentration
Rational functions model many real situations: average cost per unit, time as a function of speed, concentration of a substance. The approach is the same: translate the statement into a rational inequality and solve it with the sign method.
The average daily cost (in euros) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{800+15x}{x}$. Determine for which…
The average daily cost (in euros) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{800+15x}{x}$. Determine for which production levels the average cost will be strictly less than $25$ euros per component.
Solve the following rational inequality in which the numerator is a cubic polynomial: $\frac{x^3-8}{x^2-2x-3} \ge 0$
Solve the following rational inequality in which the numerator is a cubic polynomial:
$\frac{x^3-8}{x^2-2x-3} \ge 0$