Absolute Value Inequalities
A public swimming pool must keep its pH close to 7.4, but a small variation up or down is tolerated. How do you express that the pH must not stray more than a certain amount from that value? With absolute value, which measures distances. At the end of the page we come back to the pool.
Absolute value as distance
Solve the inequality $|x - 5| < 3$ and express the solution as an interval.
Solve the inequality $|x - 5| < 3$ and express the solution as an interval.
Find all real numbers $x$ that satisfy the inequality $|2x + 4| \geq 8$.
Find all real numbers $x$ that satisfy the inequality $|2x + 4| \geq 8$.
Solve the following modulus inequality: $|3x - 1| \leq 5$.
Solve the following modulus inequality: $|3x - 1| \leq 5$.
From a statement to the inequality
"The distance between $x$ and $7$ is less than $4$" is written directly as $|x-7|<4$. The word "distance" is the signal.
Express the following statement using an absolute value inequality: "The distance between a number $x$ and $7$ is strictly less than $4$". Then, determine the solution…
Express the following statement using an absolute value inequality: "The distance between a number $x$ and $7$ is strictly less than $4$". Then, determine the solution set.
Determine the solution set of the inequality $|-x + 6| > 2$.
Determine the solution set of the inequality $|-x + 6| > 2$.
Solve the inequality $\left| \frac{x}{3} - 2 \right| \leq 1$.
Solve the inequality $\left| \frac{x}{3} - 2 \right| \leq 1$.
Solve the inequality $5 - |x - 2| > 1$.
Solve the inequality $5 - |x - 2| > 1$.
Cases with no solution or with all of the real line as the solution
Since $|x|\geq0$ always, some inequalities are solved just by reasoning about the sign.
Justify whether the following inequalities have a solution in the set of real numbers and, if so, find it: a) $|x + 4| \leq 0$ b) $|2x - 5| < -1$
Justify whether the following inequalities have a solution in the set of real numbers and, if so, find it:
- a) $|x + 4| \leq 0$
- b) $|2x - 5| < -1$
Tolerances and margins of error
A tolerance is a measurement that must be close to an ideal value with a maximum error: $|x-\text{ideal}|\leq\text{margin}$.
In an electronic components factory, the ideal length of a chip is $12.5$ mm. For quality control reasons, a maximum margin of error of $0.15$ mm is accepted. Write a…
In an electronic components factory, the ideal length of a chip is $12.5$ mm. For quality control reasons, a maximum margin of error of $0.15$ mm is accepted. Write a modulus inequality that models this situation for the length $L$ of the chip and solve it to find the interval of acceptable lengths.
The pH of a swimming pool must be maintained at an optimal level of $7.4$. For the water to be suitable for swimming, the absolute difference between the current pH and…
The pH of a swimming pool must be maintained at an optimal level of $7.4$. For the water to be suitable for swimming, the absolute difference between the current pH and the optimal pH must not exceed $5\%$ of the optimal value. Set up the corresponding modulus inequality and determine the range of permitted pH levels.
The temperature $T$ (in degrees Celsius) of an industrial furnace fluctuates during operation such that $|2T - 300| \leq 50$. If for safety reasons in the smelting…
The temperature $T$ (in degrees Celsius) of an industrial furnace fluctuates during operation such that $|2T - 300| \leq 50$. If for safety reasons in the smelting process the temperature cannot drop below $130^\circ$C at any time, verify mathematically whether the furnace meets the safety regulations guaranteed by its fluctuation.
Quadratic expressions, rational expressions, and systems
If inside the absolute value there is a quadratic or rational expression, the method is the same: you translate it into a double inequality or into two pieces and solve each inequality with its own techniques (roots and signs). If there are several conditions at once (a system), you intersect the solutions.
When there is a sum of absolute values, like $|x-1|+|x-3|$, distance is not enough: you split the line into pieces according to where each expression changes sign (at $x=1$ and at $x=3$). The graph helps you see the result.
Solve the quadratic modulus inequality $|x^2 - 5| < 4$.
Solve the quadratic modulus inequality $|x^2 - 5| < 4$.
Find the values of $x$ that satisfy the inequality $|2x - 3| < x + 1$.
Find the values of $x$ that satisfy the inequality $|2x - 3| < x + 1$.
Solve the following system of linear modulus inequalities: $$ \begin{cases} |x - 2| \leq 4 \\ |x + 1| > 2 \end{cases} $$
Solve the following system of linear modulus inequalities:
$$ \begin{cases} |x - 2| \leq 4 \\ |x + 1| > 2 \end{cases} $$Solve the rational modulus inequality $\frac{2}{|x - 3|} \geq 1$.
Solve the rational modulus inequality $\frac{2}{|x - 3|} \geq 1$.
Find the solution set of the inequality $|x - 2| < |x + 4|$ by geometrically interpreting the modulus as the distance between points on the number line.
Find the solution set of the inequality $|x - 2| < |x + 4|$ by geometrically interpreting the modulus as the distance between points on the number line.
Solve the inequality $|x^2 - 4x| > 3x$. Express the solution as a union of intervals.
Solve the inequality $|x^2 - 4x| > 3x$. Express the solution as a union of intervals.
Solve the following system of combined inequalities: $$ \begin{cases} |x^2 - 4| \leq 5 \\ |2x - 1| \geq 3 \end{cases} $$
Solve the following system of combined inequalities:
$$ \begin{cases} |x^2 - 4| \leq 5 \\ |2x - 1| \geq 3 \end{cases} $$Find the solution set of the inequality $|x - 1| + |x - 3| \leq 6$. The figure shows the graph of $f(x) = |x-1| + |x-3|$ (in blue) and the line $y = 6$ (in green). You…
Find the solution set of the inequality $|x - 1| + |x - 3| \leq 6$.
The figure shows the graph of $f(x) = |x-1| + |x-3|$ (in blue) and the line $y = 6$ (in green). You can use it to check the result you obtain algebraically.
Solve the rational modulus inequality $\left| \frac{2x - 1}{x + 2} \right| \geq 1$.
Solve the rational modulus inequality $\left| \frac{2x - 1}{x + 2} \right| \geq 1$.
Parameters and models
Sometimes the absolute value includes a parameter, or describes a motion or a measurement. The procedure is still to translate into inequalities.
Determine for which values of the real parameter $k$ the solution set of the inequality $|x - k| \leq 3$ is strictly contained in the interval $[-1, 8]$.
Determine for which values of the real parameter $k$ the solution set of the inequality $|x - k| \leq 3$ is strictly contained in the interval $[-1, 8]$.
The speed of a vehicle at an instant $t$ (in seconds) is given by the expression $v(t) = |3t - 15|$ m/s. Determine during which time interval the speed of the vehicle is…
The speed of a vehicle at an instant $t$ (in seconds) is given by the expression $v(t) = |3t - 15|$ m/s. Determine during which time interval the speed of the vehicle is strictly less than $6$ m/s and calculate the total distance travelled in that interval, considering that the vehicle moves in a straight line.
Consider the inequality $|x^2 - a^2| \leq a^2$, where $a > 0$ is a constant real parameter. Express the solution set of this inequality in terms of the parameter $a$.
Consider the inequality $|x^2 - a^2| \leq a^2$, where $a > 0$ is a constant real parameter. Express the solution set of this inequality in terms of the parameter $a$.
In a study of optics and refraction, the deviation of a light beam from its ideal path is given by the function $d(x) = |x| - |x - 2|$. Solve the inequality…
In a study of optics and refraction, the deviation of a light beam from its ideal path is given by the function $d(x) = |x| - |x - 2|$. Solve the inequality $|d(x)| \leq 1$ to find the values of $x$ for which the deviation remains within the margins tolerable by the instrument.