Inequalities with square roots
A traffic expert examines the skid marks left by a car on a federal highway. Using the formula $v = 10\sqrt{d}$ (speed $v$ in km/h, skid length $d$ in meters), they want to know which skid lengths show that the driver was going faster than $110$ km/h. That is an inequality with a root. The key: a square root only exists if what is inside is not negative, and to remove it you square both sides, but only when you can do so without losing the inequality.
The domain first
A real square root $\sqrt{P(x)}$ only exists when $P(x) \geq 0$. Before doing any other work, we set that condition.
If the root is in the denominator, the radicand must be strictly positive, because the denominator cannot be zero. For example, $f(x) = \dfrac{1}{\sqrt{x^2 - 9}}$ requires $x^2 - 9 > 0$, that is, $x < -3$ or $x > 3$. (In the next section you will see how to solve these quadratic inequalities.)
If there are nested roots, you need the conditions for the outer one and the inner one. For $\sqrt{4 - \sqrt{x}}$ you need $x \geq 0$ (inner root) and $4 - \sqrt{x} \geq 0$, that is $\sqrt{x} \leq 4$, that is $x \leq 16$. The domain is $[0, 16]$.
Determine the domain of the real function $f(x) = \sqrt{5 - \sqrt{x - 2}}$.
Determine the domain of the real function $f(x) = \sqrt{5 - \sqrt{x - 2}}$.
Review: quadratic inequalities with a sign table
The radicand is almost always a quadratic expression, and for the domain (or when squaring) you have to solve something like $x^2 - 4x + 3 \geq 0$. Here is the method.
If the roots are not integers, they are found with the quadratic formula.
If the discriminant is negative, the parabola does not touch the $x$-axis and the expression always has the same sign. For example, $x^2 + 9 > 0$ for all $x$.
Determine the domain of the real function of a real variable $f(x) = \frac{1}{\sqrt{x^2 - 25}}$.
Determine the domain of the real function of a real variable $f(x) = \frac{1}{\sqrt{x^2 - 25}}$.
Solve the inequality $\sqrt{x^2 - 16} > 0$.
Solve the inequality $\sqrt{x^2 - 16} > 0$.
Root less than a number: $\sqrt{P(x)} < k$
When there are numbers on both sides of the root, you first isolate the root. For example, $4 - \sqrt{x+1} \leq 1$ becomes $-\sqrt{x+1} \leq -3$, and multiplying by $-1$ flips the direction: $\sqrt{x+1} \geq 3$. That is already the case in the next section.
Solve the inequality $\sqrt{x - 2} < 5$. Express the solution as an interval.
Solve the inequality $\sqrt{x - 2} < 5$. Express the solution as an interval.
Find the solution of the inequality $\sqrt{5 - 2x} \le 4$.
Find the solution of the inequality $\sqrt{5 - 2x} \le 4$.
Analyse and solve the inequality $\sqrt{2x - 7} < -3$. Justify your answer.
Analyse and solve the inequality $\sqrt{2x - 7} < -3$. Justify your answer.
Root greater than a number: $\sqrt{P(x)} > k$
In the graph, the solution of $\sqrt{3x+6} \geq 3$ is the set of $x$ where the curve is above the line $y = 3$ (or touches it).
Solve the following system of inequalities in one unknown: $\sqrt{x} \ge 2$ and $\sqrt{x} < 5$.
Solve the following system of inequalities in one unknown: $\sqrt{x} \ge 2$ and $\sqrt{x} < 5$.
Determine the solution set of the inequality $\sqrt{3x + 6} \ge 3$.
Determine the solution set of the inequality $\sqrt{3x + 6} \ge 3$.
Find the values of $x$ that satisfy the inequality $4 - \sqrt{x + 1} \le 1$.
Find the values of $x$ that satisfy the inequality $4 - \sqrt{x + 1} \le 1$.
When the other side contains the unknown
If the inequality compares $\sqrt{P(x)}$ with $g(x)$, the sign of $g(x)$ changes everything, so we split into cases.
Solve the inequality $\sqrt{x^2 - 4x + 3} \le x - 1$. Take into account the restrictions on the domain of the radicand.
Solve the inequality $\sqrt{x^2 - 4x + 3} \le x - 1$. Take into account the restrictions on the domain of the radicand.
Solve the inequality $\sqrt{x + 6} > x$. Pay special attention to cases where the right-hand side may be negative.
Solve the inequality $\sqrt{x + 6} > x$. Pay special attention to cases where the right-hand side may be negative.
Solve the inequality $\sqrt{2x^2 - 5x + 2} \ge x - 2$.
Solve the inequality $\sqrt{2x^2 - 5x + 2} \ge x - 2$.
Given the functions $f(x) = \sqrt{2x + 4}$ and $g(x) = x - 2$, solve the inequality $f(x) > g(x)$ algebraically and corroborate your result by observing the attached…
Given the functions $f(x) = \sqrt{2x + 4}$ and $g(x) = x - 2$, solve the inequality $f(x) > g(x)$ algebraically and corroborate your result by observing the attached graph.
Problems with roots
Many quantities use roots: the period of a pendulum, distances, braking speed. The method is the same: translate the statement into an inequality, fix the domain, solve, and filter with the context (a length or a time cannot be negative).
Some tips for problems:
-
If there is a root and a number on each side, isolate the root first and then apply the rules.
-
With a diagonal $d$ and a base $b$ in a rectangle, the height is $\sqrt{d^2 - b^2}$. Since $d^2 - b^2 \geq 0$, requiring a minimum height $h$ is equivalent to $d^2 \geq b^2 + h^2$.
-
With a pendulum period $T = 2\pi\sqrt{L/g}$ and $g \approx \pi^2$, it becomes $T = 2\sqrt{L}$.
-
If the unknown is a number of pieces or batches, an integer value is required: use the integer part of the endpoint.
The speed $v$ (in km/h) of a vehicle before braking sharply can be estimated using the empirical formula $v = 10\sqrt{d}$, where $d$ is the length of the skid marks in…
The speed $v$ (in km/h) of a vehicle before braking sharply can be estimated using the empirical formula $v = 10\sqrt{d}$, where $d$ is the length of the skid marks in meters. If the speed limit on a stretch of divided highway is $120\text{ km/h}$, set up an inequality and determine which skid lengths indicate that the driver was exceeding the speed limit.
The period $T$ of oscillation of a simple pendulum is given by the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum in meters and $g$ is the…
The period $T$ of oscillation of a simple pendulum is given by the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum in meters and $g$ is the acceleration due to gravity. Assuming $g \approx \pi^2 \text{ m/s}^2$, find the range of lengths $L$ for which the period of oscillation is strictly between $1$ and $3$ seconds.
A surveillance drone is programmed to remain at a distance strictly less than $5$ km from a base located at the origin of coordinates $(0,0)$. If the path of the drone…
A surveillance drone is programmed to remain at a distance strictly less than $5$ km from a base located at the origin of coordinates $(0,0)$. If the path of the drone is given by points of the form $(x, 3)$, set up and solve the radical inequality that determines the permitted values of $x$ for the path.
The production time $T$ in hours to manufacture $x$ batches of an electronic component follows the model $T(x) = 3\sqrt{2x + 1} + 2$. If the maximum daily shift is $14$…
The production time $T$ in hours to manufacture $x$ batches of an electronic component follows the model $T(x) = 3\sqrt{2x + 1} + 2$. If the maximum daily shift is $14$ hours, what is the maximum number of full batches that can be produced within the time limit?
The design of a rectangular solar panel requires its base to measure a fixed $2$ meters and its diagonal $d$ to satisfy the relationship $d = \sqrt{x^2 + 4x + 8}$…
The design of a rectangular solar panel requires its base to measure a fixed $2$ meters and its diagonal $d$ to satisfy the relationship $d = \sqrt{x^2 + 4x + 8}$ meters, where $x$ is a structural adjustment parameter. Find the values of $x$ that ensure the height of the panel is at least $2$ meters.
A telecommunications antenna has an effective coverage radius of $10\text{ km}$. A vehicle travels along a straight road and its distance to the antenna in kilometers at…
A telecommunications antenna has an effective coverage radius of $10\text{ km}$. A vehicle travels along a straight road and its distance to the antenna in kilometers at time $t$ (in minutes) is given by the function $d(t) = \sqrt{t^2 - 12t + 52}$. Determine the time interval during which the vehicle has coverage.
To connect a base station to a repeater, an optical fiber cable must be laid. The cost of laying the cable across rocky terrain is proportional to the straight-line…
To connect a base station to a repeater, an optical fiber cable must be laid. The cost of laying the cable across rocky terrain is proportional to the straight-line distance $\sqrt{x^2 + 16}$, and the cost across flat terrain is proportional to the remaining distance $(8 - x)$. The simplified total cost model is given by $C(x) = 2\sqrt{x^2 + 16} + (8 - x)$. For which values of $x \ge 0$ does the total cost satisfy the budget constraint $C(x) \le 16$?
In special relativity theory, the Lorentz time dilation factor is given by $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the speed of the object and $c$…
In special relativity theory, the Lorentz time dilation factor is given by $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the speed of the object and $c$ is the speed of light. Set up and solve an inequality to find the range of speeds $v$ (expressed as a fraction of $c$, assuming $v \ge 0$) for which the Lorentz factor is strictly greater than $2$.
Systems, parameters, and nested roots
Systems. Solve each inequality separately and intersect the results.
Parameter. Requiring that $\sqrt{x^2 - mx + 9} > 0$ hold for all $x$ is equivalent to requiring that the radicand be strictly positive for all $x$. That happens when the parabola does not touch the $x$-axis: the discriminant is negative, $m^2 - 36 < 0$, that is $-6 < m < 6$. With $m = \pm 6$ the radicand equals $0$ at one point and the root is not $>0$ there.
Move $m$: when the graph breaks apart (there are values of $x$ with no curve) the radicand becomes negative on some interval.
Nested roots. Work from the outside in, but pay attention to the domain of each one. For $\sqrt{x - \sqrt{1-x}} \geq 1$:
Find the set of solutions that simultaneously satisfy the following system of inequalities: $\sqrt{3x + 1} \le 4$ $x^2 - 2x - 3 > 0$
Find the set of solutions that simultaneously satisfy the following system of inequalities:
$\sqrt{3x + 1} \le 4$
$x^2 - 2x - 3 > 0$
Determine the values of the parameter $m$ for which the inequality $\sqrt{x^2 - mx + 9} > 0$ is satisfied for any real number $x$.
Determine the values of the parameter $m$ for which the inequality $\sqrt{x^2 - mx + 9} > 0$ is satisfied for any real number $x$.
Solve the inequality with nested radicals $\sqrt{x - \sqrt{1 - x}} \ge 1$.
Solve the inequality with nested radicals $\sqrt{x - \sqrt{1 - x}} \ge 1$.
Summary
Back to the start: with $10\sqrt{d} > 110$ you get $\sqrt{d} > 11$, and therefore $d > 121$ m.