Inequalities with square roots
A real square root only exists if the expression inside it is not negative. This single restriction is the key to the whole section: before doing any calculation, you must fix the domain of the inequality. Then, to "remove" the root, you square both sides, but with an extra condition on the sign of the other side, or you will generate solutions that are not real.
The domain first: the radicand cannot be negative
Solve the inequality $\sqrt{x^2 - 16} > 0$.
Solve the inequality $\sqrt{x^2 - 16} > 0$.
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 $f(x) = \sqrt{5 - \sqrt{x - 2}}$.
Solve the inequality with nested radicals $\sqrt{x - \sqrt{1 - x}} \ge 1$.
Solve the inequality with nested radicals $\sqrt{x - \sqrt{1 - x}} \ge 1$.
Root less than a number: $\sqrt{P(x)}<k$
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: $\sqrt{P(x)}>k$
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 $4 - \sqrt{x + 1} \le 1$.
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 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.
When the other side contains the unknown
If the inequality is $\sqrt{P(x)}\lessgtr g(x)$ with $g(x)$ depending on $x$, looking at a fixed number is no longer enough: you must discuss the sign of $g(x)$, because the squaring step is only valid without conditions when both sides have a known sign.
To handle a cost model such as $2\sqrt{x^2+9}+(6-x)\leq 12$, first isolate the root: $2\sqrt{x^2+9}\leq 6+x$, then check the sign of the right-hand side before squaring.
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:…
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.
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.
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…
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 $C(x) \le 16$?
Given the inequality with two square roots $\sqrt{2x + 3} - \sqrt{x - 1} \leq 2$: a) Determine the joint domain of existence of the inequality, explaining why the…
Given the inequality with two square roots $\sqrt{2x + 3} - \sqrt{x - 1} \leq 2$:
- a) Determine the joint domain of existence of the inequality, explaining why the condition must apply to both radicands simultaneously.
- 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.
- 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).
Problems and systems with roots
Many physical quantities are expressed with a root (the period of a pendulum, distance, braking speed on an Irish motorway).
The approach is the same: identify the domain, translate the problem statement and solve using the rules above. It can also be combined with another inequality in a system.
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…
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.
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 $x$.