Exercises: Square root inequalities

Solve the inequality $\sqrt{x - 2} < 5$. Express the solution as an interval.

Determine the solution set of the inequality $\sqrt{3x + 6} \ge 3$.

Find the values of $x$ that satisfy the inequality $4 - \sqrt{x + 1} \le 1$.

Solve the inequality $\sqrt{x^2 - 16} > 0$.

Find the solution of the inequality $\sqrt{5 - 2x} \le 4$.

Solve the following system of inequalities in one unknown: $\sqrt{x} \ge 2$ and $\sqrt{x} < 5$.

Determine the domain of the real function of a real variable $f(x) = \frac{1}{\sqrt{x^2 - 25}}$.

Analyse and solve the inequality $\sqrt{2x - 7} < -3$. Justify your answer.

Cian tries to solve the irrational inequality $\sqrt{2x - 8} \leq -2$. To do this, he writes the following steps on his exam:

1) "I square both sides: $(\sqrt{2x - 8})^2 \leq (-2)^2 \implies 2x - 8 \leq 4$".

2) "I solve for the unknown: $2x \leq 12 \implies x \leq 6$".

3) "I find the domain of the square root: $2x - 8 \geq 0 \implies x \geq 4$".

4) "I intersect both conditions and the solution is the interval $[4, 6]$".

Explain in detail what serious conceptual error Cian made in step 1, why the property of squaring both sides is not applicable in this case, and what the real, justified solution to the inequality is without needing to do any calculations.

Consider the inequality $\sqrt{25 - x} \leq 3$.

1. a) Before solving it algebraically, reason qualitatively what type of interval you expect: bounded on the left, on the right, or at both ends? Justify it by analyzing what happens to the radicand as $x$ increases and what restriction the existence of the square root imposes.
2. b) Make a mental estimate by evaluating simple values ($x = 0$, $x = 16$, $x = 25$) to conjecture whether the solution will contain values close to $25$ or close to $0$.
3. c) Solve the inequality using the formal algebraic procedure (domain and squaring both sides) and check whether your initial estimate was consistent.

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 metres 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 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.

Solve the inequality $\sqrt{x^2 - 4x + 3} \le x - 1$. Take into account the restrictions on the domain of the radicand.

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?

Determine the domain of the real function $f(x) = \sqrt{5 - \sqrt{x - 2}}$.

The design of a rectangular solar panel requires its base to measure a fixed $2$ metres and its diagonal $d$ to satisfy the relationship $d = \sqrt{x^2 + 4x + 8}$ metres, where $x$ is a structural adjustment parameter. Find the values of $x$ that ensure the height of the panel is at least $2$ metres.

Find the set of solutions that simultaneously satisfy the following system of inequalities:

$\sqrt{3x + 1} \le 4$

$x^2 - 2x - 3 > 0$

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 metres. If the speed limit on a stretch of dual carriageway is $120\text{ km/h}$, set up an inequality and determine which skid lengths indicate that the driver was exceeding the speed limit.

The Road Safety Authority (RSA) and accident investigation teams use mathematical models to estimate speed prior to braking based on tire skid marks on asphalt. On a wet stretch of an Irish regional road, the estimated speed $v$ (in $\text{km/h}$) and the braking distance $d$ (in meters) are related by the expression $v = \sqrt{120 \times d}$.

Following an incident, a driver claims they were driving within the legal speed limit (that is, $v \leq 90\text{ km/h}$), while measurements from an earlier average speed radar confirm that their speed was not lower than $60\text{ km/h}$.

1. a) Translate this situation into a double inequality with square roots in terms of distance $d$.
2. b) Determine the domain of physical and mathematical validity of the expression.
3. c) Solve the inequality to find the permissible interval of braking lengths $d$ that would make both testimonies compatible, expressing the result as an interval with decimal numbers rounded to the hundredths.

You want to construct an inequality of the form $\sqrt{a \times x + b} < c$, where $a$, $b$, and $c$ are fixed real numbers, such that its solution set is exactly the half-open interval $[-3, 5)$.

1. a) Justify, using the properties of square roots, why the constant $c$ must be strictly positive.
2. b) Knowing that $a = 2$, determine in a reasoned manner the values that the parameters $b$ and $c$ must take (requiring that $c$ be an integer).
3. c) Explain why the left end of the solution interval is closed ($-3 \in [-3, 5)$) while the right end is open ($5 \notin [-3, 5)$), relating each of them to a different stage of the solving process.

Solve the inequality $\sqrt{x + 6} > x$. Pay special attention to cases where the right-hand side may be negative.

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 kilometres 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.

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 $\sqrt{2x^2 - 5x + 2} \ge x - 2$.

To connect a base station to a repeater, an optical fibre 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 $C(x) \le 16$?

Solve the inequality with nested radicals $\sqrt{x - \sqrt{1 - x}} \ge 1$.

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 $2$.

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.

To solve the inequality $\sqrt{2x + 5} > x - 5$, two classmates propose different strategies:

• Aoife proposes squaring both sides directly: $2x + 5 > (x - 5)^2$, solving the resulting polynomial inequality, and intersecting the solution set with the domain of existence of the square root ($2x + 5 \geq 0$).

• Seán argues that Aoife's method is incorrect because the sign of $x - 5$ is not constant. His method consists of distinguishing two cases: when $x - 5 < 0$ (where the inequality is straightforward if the root exists) and when $x - 5 \geq 0$ (where it is valid to square both sides), finally taking the union of the solutions of both cases.

1. a) Show by means of a numerical counterexample (choosing a specific value of $x$) that Aoife's method omits valid solutions.
2. b) Formally justify why Seán's method is rigorous based on the sign of each side and the monotonicity of the function $f(t) = t^2$.
3. c) Fully develop Seán's method and find the final solution interval of the inequality.

Given the inequality with two square roots $\sqrt{2x + 3} - \sqrt{x - 1} \leq 2$:

1. a) Determine the joint domain of existence of the inequality, explaining why the condition must apply to both radicands simultaneously.
2. b) Explain why rearranging the inequality to $\sqrt{2x + 3} \leq 2 + \sqrt{x - 1}$ before squaring is an essential step to ensure that both sides are non-negative and preserve the equivalence of the inequality.
3. c) Solve the inequality by applying the necessary transformations with justification and express the final set of solutions as a closed interval whose endpoints are exact real numbers (in terms of irreducible radicals).