Quadratic Inequalities
A corner store sells passion fruit juice and its daily profit depends on how many units it prepares. With few units it loses money, with a medium amount it makes money, and with too many it loses money again. Between which amounts does it make a profit? That question is a second-degree inequality: you don't "solve for x," you study the sign of a parabola.
The method: roots and signs by intervals
The graphical method says the same thing using the parabola: $f(x)>0$ where the graph is above the x-axis, and $f(x)<0$ where it is below. The roots are the points where it crosses the axis.
Solve the following quadratic inequality and express the solution in interval notation: $x^2 - 5x + 6 \ge 0$
Solve the following quadratic inequality and express the solution in interval notation:
$x^2 - 5x + 6 \ge 0$
Solve the incomplete quadratic inequality: $2x^2 - 8x \le 0$
Solve the incomplete quadratic inequality:
$2x^2 - 8x \le 0$
Find the solution set of the inequality: $-x^2 + 4x + 5 > 0$
Find the solution set of the inequality:
$-x^2 + 4x + 5 > 0$
Inverted parabola and general form
When $a<0$ the parabola opens downward: it is positive between the roots and negative outside them. To avoid mistakes, you can multiply the whole inequality by $-1$ and reverse the direction of the inequality sign.
If the inequality comes with products, expand and simplify first until you reach the general form. For example, $(x-1)(x+2) Determine the values of $x$ that satisfy the following inequality: $3x^2 - 27 < 0$ Solve the following inequality by first expanding and simplifying it to its general form, and express the result as an interval: $(2x - 1)^2 - (x + 3)(x - 3) < 5x + 14$Determine the values of $x$ that satisfy the following inequality: $3x^2 - 27 < 0$
Solve the following inequality by first expanding and simplifying it to its general form, and express the result as an interval: $(2x - 1)^2 - (x + 3)(x - 3) < 5x + 14$
Cases without two roots: double root and negative discriminant
There are not always two distinct roots. Look at the discriminant $\Delta=b^2-4ac$:
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If $\Delta<0$, the parabola does not cross the x-axis: it has the same sign on the whole line, the sign of $a$ (the roots are complex conjugates, not real).
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If $\Delta=0$, there is a double root: the polynomial is a perfect square, it is never negative (if $a>0$) and equals $0$ only at the root.
Find the solution set of the inequality: $x^2 + 6x + 9 \le 0$
Find the solution set of the inequality:
$x^2 + 6x + 9 \le 0$
State the solution set of the inequality, giving reasons for your answer: $x^2 - 2x + 5 > 0$
State the solution set of the inequality, giving reasons for your answer:
$x^2 - 2x + 5 > 0$
Determine for which values of $x$ the following inequality holds: $4x^2 - 12x + 9 > 0$
Determine for which values of $x$ the following inequality holds:
$4x^2 - 12x + 9 > 0$
Reading graphs and domains
If you are given the graph of $y=f(x)$, you don't need to calculate: $f(x)\le 0$ is read where the curve is on or below the x-axis. Look at the crossing points (roots) and decide whether the endpoints are included (with $\le$ or $\ge$) or not.
A very common use is the domain of a square root: $\sqrt{f(x)}$ exists only when $f(x)\ge 0$. And if there is a root in a denominator, you need $f(x)>0$ (it cannot be zero).
From the graphical representation of the parabola $y = f(x)$, visually determine the solution set of the inequality $f(x) \le 0$.
From the graphical representation of the parabola $y = f(x)$, visually determine the solution set of the inequality $f(x) \le 0$.
Find the domain of definition of the following real function of a real variable: $f(x) = \sqrt{-2x^2 + 7x - 3}$
Find the domain of definition of the following real function of a real variable:
$f(x) = \sqrt{-2x^2 + 7x - 3}$
Find the domain of definition of the following function, which combines restrictions from radicands and denominators: $g(x) = \frac{1}{\sqrt{x^2 - 4}} + \sqrt{9 - x^2}$
Find the domain of definition of the following function, which combines restrictions from radicands and denominators:
$g(x) = \frac{1}{\sqrt{x^2 - 4}} + \sqrt{9 - x^2}$
Systems of inequalities in one variable
A system requires all of its inequalities to hold at the same time. You solve each one separately and intersect the solutions (the common points).
Drawing both solutions on parallel number lines and looking at where they overlap helps avoid mistakes. An inequality with absolute value such as $|x^2-4x|\le 5$ becomes a system: $-5\le x^2-4x\le 5$, that is, $x^2-4x\le 5$ and $x^2-4x\ge -5$.
Solve the following system of quadratic inequalities in one variable: $x^2 - 4 \le 0$ $x^2 - x \ge 0$
Solve the following system of quadratic inequalities in one variable:
$x^2 - 4 \le 0$
$x^2 - x \ge 0$
Solve the following system of inequalities and give the final solution as an interval: $x^2 - 3x - 10 < 0$ $2x + 5 \ge 1$
Solve the following system of inequalities and give the final solution as an interval:
$x^2 - 3x - 10 < 0$
$2x + 5 \ge 1$
Solve the modulus inequality by first transforming it into a system of quadratic inequalities: $|x^2 - 4x| \le 5$
Solve the modulus inequality by first transforming it into a system of quadratic inequalities:
$|x^2 - 4x| \le 5$
Problems with quadratic inequalities
In problems, the inequality comes from a quadratic function that models something real: the height of a ball, the profit of a business, the area of a lot. Steps: define the unknown, set up the inequality, solve it and interpret the solution (discard meaningless values, such as negative lengths).
The height $h$ in meters of a projectile launched vertically upward as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Set up and…
The height $h$ in meters of a projectile launched vertically upward as a function of time $t$ in seconds is given by the expression $h(t) = -5t^2 + 20t + 2$. Set up and solve the corresponding inequality to determine during what time interval the projectile is at a height strictly greater than $17$ meters.
The daily profit of a company, expressed in hundreds of pesos, when manufacturing $x$ units of a product is modeled by the quadratic function…
The daily profit of a company, expressed in hundreds of pesos, when manufacturing $x$ units of a product is modeled by the quadratic function $B(x) = -30x^2 + 2\,400x - 21\,000$. How many units must it manufacture daily in order not to make a loss? Express the result as an interval.
A $40$-meter rope is available to fence off a rectangular plot, using an existing wall as one of the sides. If $x$ is the length of the sides perpendicular to the wall,…
A $40$-meter rope is available to fence off a rectangular plot, using an existing wall as one of the sides. If $x$ is the length of the sides perpendicular to the wall, for what values of $x$ will the enclosed area be at least $150 \text{ m}^2$? Note that the dimensions of the plot must be positive.
The area of the central rectangular zone of a sports court must be strictly less than $800 \text{ m}^2$. If the length of the court is twice its width $x$ plus $10$…
The area of the central rectangular zone of a sports court must be strictly less than $800 \text{ m}^2$. If the length of the court is twice its width $x$ plus $10$ meters, set up the corresponding inequality and deduce what values the width of the court can take.
The population of a certain invasive species in a lake, measured in thousands of individuals, follows the model $P(t) = -0.2t^2 + 3.2t + 4$, where $t$ is the time in…
The population of a certain invasive species in a lake, measured in thousands of individuals, follows the model $P(t) = -0.2t^2 + 3.2t + 4$, where $t$ is the time in months since the invasion was detected. The authorities decide to intervene when the population exceeds $14,000$ individuals. However, the ecological treatment is only effective if it can be applied for at least $3$ consecutive months during which the population remains above this threshold. Set up the inequality and determine whether it will be possible to successfully apply the treatment according to this model.
An open-topped tray is to be constructed from a rectangular sheet of aluminum measuring $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ from the…
An open-topped tray is to be constructed from a rectangular sheet of aluminum measuring $30 \times 20 \text{ cm}$, by cutting out identical squares of side $x$ from the four corners and folding up the resulting flaps. If design requirements dictate that the base area of the tray must be at least $200 \text{ cm}^2$, set up and solve the corresponding inequality to find the interval of possible values of $x$.
The main suspension cable of a bridge takes the shape of a parabola. Taking the lowest point of the cable as the origin of coordinates, the height $y$ (in meters) of the…
The main suspension cable of a bridge takes the shape of a parabola. Taking the lowest point of the cable as the origin of coordinates, the height $y$ (in meters) of the cable above the roadway at a horizontal distance $x$ (in meters) from the center is given by the function $y = 0.005x^2 + 2$. For aviation safety reasons, obstruction lights must be installed on the sections of the cable whose height above the roadway is between $10 \text{ m}$ and $22 \text{ m}$. Set up the system of inequalities and calculate the intervals of horizontal distance from the center where these lights will be installed.
Comparing two functions and changing variables
To find out when $f(x) A biquadratic inequality such as $x^4-13x^2+36\le 0$ is solved with $u=x^2$: it becomes $u^2-13u+36\le0$, with roots $u=4$ and $u=9$, and solution $4\le u\le 9$. You undo the substitution: $4\le x^2\le 9$, which gives $[-3,-2]\cup[2,3]$. Given the quadratic functions $f(x) = x^2 - 2x - 3$ and $g(x) = -x^2 + 4x + 5$, determine analytically on which interval of the number line $f(x) < g(x)$ holds. Based on the attached graph representing both functions and the area enclosed between them, verify the consistency of your result. Solve the following biquadratic inequality using a substitution of variable and express the final solution as a union of intervals: $x^4 - 13x^2 + 36 \le 0$Given the quadratic functions $f(x) = x^2 - 2x - 3$ and $g(x) = -x^2 + 4x + 5$, determine analytically on which interval of the number line $f(x) < g(x)$ holds. Based on…
Solve the following biquadratic inequality using a substitution of variable and express the final solution as a union of intervals: $x^4 - 13x^2 + 36 \le 0$
With a parameter
Sometimes you are asked for which values of a parameter ($m$, $k$…) the inequality holds for every real $x$. This happens when the parabola does not cross the x-axis and has the right sign.
Determine for which values of the parameter $m$ the quadratic inequality $x^2 - mx + (m + 3) > 0$ holds for any real number $x$.
Determine for which values of the parameter $m$ the quadratic inequality $x^2 - mx + (m + 3) > 0$ holds for any real number $x$.