Equations · Inequalities and systems of inequalities
Rational inequalities
A corner store wants to know how many units of a product it should prepare so that the average cost per unit falls below a certain value. The average cost is a fraction where the unknown is in the denominator. The temptation is to "get rid of" the denominator by cross-multiplying, but that is dangerous: you do not know whether what you are multiplying by is positive or negative. The correct method is to study the sign of the numerator and the denominator separately, using a sign chart.
Why we do not cross-multiply
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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:
When the numerator or the denominator has a higher degree, we first factor it (difference of squares, trinomials, synthetic division) and look for all the roots. Each factor contributes a critical point.
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Find the solution set of the inequality: $\frac{2x+6}{x-1} \le 0$
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 reduce to a single fraction. We never cross-multiply.
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Find the solution set of the inequality: $\frac{x+3}{x-2} \ge 2$
A square root exists only if what is inside is $\geq 0$. If what is inside is a fraction, the domain comes from a rational inequality.
For a system of rational inequalities, we solve each one separately and intersect the solutions: only the $x$ values that satisfy all of them at once work.
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Determine the domain of definition of the function of a real variable $f(x) = \sqrt{\frac{x^2-x-2}{x^2+4x+3}}$.
Determine the domain of definition of the function of a real variable $f(x) = \sqrt{\frac{x^2-x-2}{x^2+4x+3}}$.
Sometimes the question is different: for which values of a parameter $k$ does the inequality hold for every real number $x$? Here we do not solve for $x$: we analyze the sign of each polynomial.
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Determine for which values of the real parameter $k$ the following inequality holds for all $x \in \mathbb{R}$: $\frac{x^2-kx+9}{x^2+1} > 0$
Determine for which values of the real parameter $k$ the following inequality holds for all $x \in \mathbb{R}$:
Rational functions model average cost per unit, time as a function of speed, or the concentration of a medication. The statement is translated into a rational inequality and solved with the sign method, remembering that the variable in the context is usually positive.
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The average daily cost (in pesos) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{48\,000+900x}{x}$. Determine for which…
The average daily cost (in pesos) of producing $x$ electronic components in a factory is given by the expression $C(x) = \frac{48\,000+900x}{x}$. Determine for which production levels the average cost will be strictly less than $1\,500$ pesos per component.
The time $T$ (in hours) it takes a boat to travel a distance of $120$ km downstream is modelled by $T(v) = \frac{120}{v+4}$, where $v$ is the boat's own speed in km/h.…
The time $T$ (in hours) it takes a boat to travel a distance of $120$ km downstream is modelled by $T(v) = \frac{120}{v+4}$, where $v$ is the boat's own speed in km/h. For which speeds $v > 0$ will the journey take less than $5$ hours?
The concentration of a drug in the bloodstream, measured in mg/L, $t$ hours after its administration is given by the function $C(t) = \frac{4t}{t^2+4}$. During which…
The concentration of a drug in the bloodstream, measured in mg/L, $t$ hours after its administration is given by the function $C(t) = \frac{4t}{t^2+4}$. During which time interval $t \ge 0$ is the concentration of the drug greater than or equal to $1$ mg/L?