Exercises: Inequalities with square roots

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 with one unknown: $\sqrt{x} \ge 2$ and $\sqrt{x} < 5$.

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

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

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 feet and $g$ is the acceleration due to gravity. Assuming $g \approx \pi^2 \text{ ft/s}^2$, 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 stay at a distance strictly less than $5$ miles from a base located at the origin of coordinates $(0,0)$. If the drone's trajectory is given by points of the form $(x, 3)$, set up and solve the inequality with roots that determines the allowed values of $x$ for the trajectory.

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 work shift is $14$ hours, what is the maximum number of complete batches that can be produced while respecting 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 $2$ feet and its diagonal $d$ to satisfy the relationship $d = \sqrt{x^2 + 4x + 8}$ feet, where $x$ is a structural adjustment parameter. Find the values of $x$ that guarantee the height of the panel is at least $2$ feet.

Find the solution set that simultaneously satisfies the following system of inequalities:

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

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

The speed $v$ (in mph) 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 feet. If the speed limit on a stretch of highway is $120\text{ mph}$, set up an inequality and determine what lengths of skid marks indicate that the driver was exceeding the speed limit.

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

A telecommunications antenna has an effective coverage radius of $10\text{ mi}$. A vehicle travels along a straight road and its distance to the antenna in miles 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, a fiber optic cable must be laid. The cost of laying the cable over rocky terrain is proportional to the straight-line distance $\sqrt{x^2 + 16}$, and the cost over 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 what values of $x \ge 0$ does the total cost meet the budget constraint $C(x) \le 16$?

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

In the special theory of relativity, 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 graphical representation.