Inequalities with square roots
After a skid on the Autopista Duarte, the traffic police estimate the car's speed with the formula $v = 10\sqrt{d}$, where $d$ is the length of the skid marks in meters. If the limit is 120 km/h, how long does the skid have to be to know the driver was going faster? To answer, you need to solve an inequality with a root. A real square root only exists if what is inside is not negative, and that idea is the key to this whole section.
The domain first
Sometimes there are several conditions at once, for example with nested roots.
There are also denominators: in $\dfrac{1}{\sqrt{P(x)}}$ the radicand must be strictly positive, because the root cannot equal 0 in a denominator. And in general, $\sqrt{P(x)}>0$ is equivalent to $P(x)>0$.
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 $f(x) = \frac{1}{\sqrt{x^2 - 25}}$.
Determine the domain of the real function $f(x) = \sqrt{5 - \sqrt{x - 2}}$.
Determine the domain of the real function $f(x) = \sqrt{5 - \sqrt{x - 2}}$.
Root less than a number
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
If the root is not alone, you isolate it first. For example, $4-\sqrt{x+1}\leq 1$ becomes $\sqrt{x+1}\geq 3$, which gives $x+1\geq 9$, that is, $x\geq 8$ (and this already satisfies the domain $x\geq -1$).
A system is solved by intersecting the solutions. For $\sqrt{x}\geq 2$ and $\sqrt{x}<5$: the first gives $x\geq 4$, the second gives $0\leq x<25$, and the common solution is $[4, 25)$.
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 $4 - \sqrt{x + 1} \le 1$.
Solve the inequality $\sqrt{x^2 - 16} > 0$.
Solve the inequality $\sqrt{x^2 - 16} > 0$.
Root versus an expression with $x$
When the inequality is $\sqrt{P(x)}\lessgtr g(x)$ and $g(x)$ contains $x$, the sign of $g(x)$ changes depending on the value of $x$. You have to separate 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.
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.
Quadratic radicand and parameters
Many times the radicand is a quadratic. Then the domain comes from a second-degree inequality: you find the roots and study the sign by intervals.
When there is a system with a root and a quadratic, you solve each inequality separately and intersect them. For example, $\sqrt{3x+1}\leq 4$ gives $-\tfrac{1}{3}\leq x\leq 5$, and $x^2-2x-3>0$ gives $x<-1$ or $x>3$. The common solution is $(3, 5]$.
With a parameter $m$: $\sqrt{x^2-mx+9}>0$ is equivalent to $x^2-mx+9>0$. For it to hold for every real $x$, the parabola must not cross the $x$-axis: negative discriminant, $m^2-36<0$, that is, $-6 For a root like $\sqrt{x^2+4x+8}$, the discriminant is $16-32<0$, so the radicand is always positive and the domain is all of $\mathbb{R}$. Find the set of solutions that simultaneously satisfy the following system of inequalities: $\sqrt{3x + 1} \le 4$ $x^2 - 2x - 3 > 0$ Solve the inequality $\sqrt{2x^2 - 5x + 2} \ge x - 2$. Determine the values of the parameter $m$ for which the inequality $\sqrt{x^2 - mx + 9} > 0$ is satisfied for any real number $x$.Find the set of solutions that simultaneously satisfy the following system of inequalities: $\sqrt{3x + 1} \le 4$ $x^2 - 2x - 3 > 0$
Solve the inequality $\sqrt{2x^2 - 5x + 2} \ge x - 2$.
Determine the values of the parameter $m$ for which the inequality $\sqrt{x^2 - mx + 9} > 0$ is satisfied for any real number $x$.
Problems with roots
Many quantities are expressed with roots: the period of a pendulum, a distance, the speed from a skid. The method is always the same: translate the statement into an inequality, find the domain, solve, and check that the solution makes physical sense.
For the distance from the origin to the point $(x, 3)$, the distance is $\sqrt{x^2+9}$. The condition "less than 5" is $\sqrt{x^2+9}<5$, which gives $x^2<16$, that is, $(-4, 4)$.
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 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 highway is $120\text{ km/h}$, set up an inequality and determine which skid lengths indicate that the driver was exceeding the speed limit.
Challenges with models
Here you combine everything: isolating the root, checking the domain, and studying the sign.
When there is a root and a linear term (as in $2\sqrt{x^2+16}+(8-x)\leq 16$), isolate the root first, $2\sqrt{x^2+16}\leq 8+x$, and check the sign of the right side: for $x\geq 0$ it is positive, so you can square, $4x^2+64\leq x^2+16x+64$, which gives $3x^2-16x\leq 0$ and $0\leq x\leq \tfrac{16}{3}$.
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 $C(x) \le 16$?
Back to the beginning: the skid that reveals speeding is the one longer than 144 m.