Equations · Inequalities and systems of inequalities
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 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.
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.
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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.
Many physical quantities are expressed with a root (the period of a pendulum, distance, braking speed). 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.
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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.