Exercises: Modulus

Calculate the numerical value of the following expressions using the definition of modulus:

1. a) $|-12.5|$

1. b) $|7 - 15|$

1. c) $|\pi - 4|$

1. d) $|1 - \sqrt{2}|$

The temperature in a town in the Irish Midlands at 6:00 a.m. was $-3\,^\circ C$, while at 3:00 p.m. it rose to $14\,^\circ C$. Use the concept of absolute value to express and calculate the temperature distance (the difference in degrees) between the two measurements.

In the figure you can see the two temperatures placed on the number line of degrees.

1. a) Write, using absolute value, the expression that gives the distance between the two temperatures.

1. b) Calculate its value and interpret it in the context of the problem.

Find the values of $x$ that satisfy the following equalities:

1. a) $|x| = 8$

1. b) $|x - 3| = 5$

1. c) $|x + 2| = 0$

Express the following sets of real numbers represented on the number line using modulus notation:

1. a) The numbers whose distance from the origin is less than $4$.

1. b) The numbers whose distance from the point $2$ is exactly equal to $3$.

Represent on the real number line and write in interval form the values of $x$ that satisfy the following inequality:

You can do it on paper and send a photo with your answer.

Match each of the following modulus expressions with its corresponding interval:

1. $|x| < 3$

2. $|x - 1| \le 2$

3. $|x + 2| < 1$

Intervals: $A = [-1, 3] \quad B = (-3, 3) \quad C = (-3, -1)$

Solve the inequality $|x| > 5$ and represent the solution graphically. Remember that this type of expression gives rise to a union of two infinite rays.

En una fábrica de componentes electrónicos, una resistencia debe tener un valor nominal de $220\ \Omega$. Por control de calidad, se acepta que el valor real $x$ de la resistencia varíe respecto al nominal en un máximo de $5\ \Omega$.

En la recta de la figura se marca el valor nominal y un valor real cualquiera $x$, junto con su distancia al valor nominal.

1. a) Escribe esta condición utilizando una desigualdad con valor absoluto.

1. b) Resuelve la desigualdad y determina el intervalo de valores óhmicos aceptables para estas resistencias.

Cian is solving the equation $|2x - 8| = -6$ on his exam and writes the following reasoning:

"By the properties of absolute value, I split into two cases:

• Case 1: $2x - 8 = -6 \Rightarrow 2x = 2 \Rightarrow x = 1$

• Case 2: $2x - 8 = 6 \Rightarrow 2x = 14 \Rightarrow x = 7$

Therefore, the equation has two solutions: $x = 1$ and $x = 7$".

1. a) Explain what fundamental conceptual error Cian has made without needing to carry out any intermediate operations.
2. b) Substitute the values $x = 1$ and $x = 7$ into the original equation to demonstrate numerically why they fail.
3. c) State the real set of solutions of the given equation and state the general condition that $k$ must satisfy for the equation $|A| = k$ to have a solution.

In an electronic components laboratory, it is established that a part works normally only if its operating temperature $x$, measured in degrees Celsius, belongs to the open interval $(-1.5;\, 6.5)$.

1. a) Calculate the midpoint (nominal value) $a$ and the radius (maximum allowable tolerance) $r$ of this interval.
2. b) Write an absolute value inequality of the form $|x - a| < r$ whose solution set is exactly that temperature interval.
3. c) Express the result using neighbourhood notation $E(a, r)$ and reason, using the interpretation of absolute value as distance on the real number line, whether a component reaching $x = 7^\circ\text{C}$ is within or outside the allowed margin.

Solve the following absolute value inequality and represent the solution on the real number line. Express the final result using interval notation:

An electronic component manufacturing company produces resistors that must have a nominal value of $220 \Omega$. For quality control reasons, a margin of error (tolerance) of $5 \Omega$ is accepted.

1. a) Write an inequality using modulus describing the acceptable resistance values $x$.
2. b) Determine the interval of allowed values so that the component is not rejected.

Calculate the values of $x$ that satisfy the following equation:

Express using a modulus inequality the set of real numbers represented in the following graph:

Find the interval resulting from the following combined condition and represent it graphically:

In a statistical study on the monthly leisure expenditure of young people in a city, it has been determined that the mean is $45$ € with a maximum error of $12$ €.

1. a) Express this situation using a neighbourhood $E(a, r)$.
2. b) Write the corresponding modulus inequality.
3. c) Could a young person spending $32$ € per month be included in this study group? Justify your answer.

Define the following absolute value function as a piecewise function:

State the expression of each branch and the interval on which each one is valid.

Determine all real numbers $x$ that lie at a distance greater than $5$ units from the point $-1$. Express the solution in three different ways:

1. As a modulus inequality.

2. As the union of two intervals.

3. Graphically on the number line.

A cooperative in Kilkenny bottles extra virgin olive oil in carboys with a nominal volume of $5\text{ l}$. Quality control regulations accept a carboy as suitable if the absolute error with respect to the nominal value does not exceed $0.8\,\%$ of said volume.

1. a) Calculate the tolerance margin $r$ in litres and set up an absolute value inequality of the form $|V - 5| \le r$ that models the acceptance condition for the actual volume $V$.
2. b) Solve the previous inequality and determine the interval of acceptable volumes in litres.
3. c) During an inspection, two carboys are taken with measured volumes of $4.955\text{ l}$ and $5.038\text{ l}$. Without solving the inequality again, use exclusively the concept of distance on the real number line to give a reasoned justification of whether each of the two carboys passes quality control or is rejected.

Consider the absolute value inequality $|x - \sqrt{5}| \le 3$.

1. a) Before operating analytically, make a reasoned estimate: knowing that $2 < \sqrt{5} < 3$ (approximately $\sqrt{5} \approx 2.2$), estimate mentally between which two integers the solution set must lie. Do you expect this set to contain the value $x = 0$? And the value $x = 5.5$? Justify your estimates based on the distance to $\sqrt{5}$ on the real number line.
2. b) Solve the inequality formally working with the exact value $\sqrt{5}$ and write the solution set in the form of a closed interval.
3. c) Use a decimal approximation of the endpoints of the interval to check whether your estimates regarding $0$ and $5.5$ were correct.

Determine the set of real numbers $x$ that simultaneously satisfy the following two distance conditions on the real number line. Express the final result using interval notation only:

1. The distance between $x$ and $2$ is less than or equal to $5$.

2. The distance between $x$ and $-1$ is strictly greater than $2$.

On the line in the figure, the two reference points, $-1$ and $2$, have been marked.

In a quality control check at an electronics components company, a $1500 \Omega$ resistor is accepted as valid if its actual value $R$ does not differ from the nominal value by more than $2\%$.

1. a) Write an inequality using modulus that represents the acceptable values of $R$.
2. b) Determine the interval of values of $R$ that would cause the component to be rejected due to being above the limit.

Solve the following equation with nested absolute values and represent the solutions on the real number line:

You can use the following number line as support, on which you must mark the solutions you obtain:

Express using a single modulus inequality the set of points represented by the union of the following two intervals:

A financial analyst observes that the price of a cryptocurrency has fluctuated following the rule $|P - 450| < 75$, where $P$ is the price in euros. At the same time, another source indicates that for security reasons, the investment is only recommended if $|P - 500| > 50$.

Determine the price interval in which both conditions are satisfied to make a safe investment according to these data.

Given the following inequalities, identify which of them defines a neighborhood with center $a = -2$ and radius $r = 3$:

1. a) $|x - 2| < 3$

1. b) $|x + 2| < 3$

1. c) $|x - 3| < 2$

Once you have identified it, write the complement of that neighborhood in the real line as a union of intervals. Also explain how you would represent it on the line: which endpoints are closed or open and in which direction each part extends.

Solve the following inequality and express the solution as a union of intervals:

Also find which natural numbers less than 10 do not belong to this solution set.

Consider the points $A$ and $B$ on the number line such that their coordinates are the solutions to the equation $|x + 4| = 6$.

Let $P$ be any point on the number line with coordinate $x$. Write an expression with modulus representing the condition: "The distance from $P$ to the midpoint of the segment $AB$ is less than 2 units". Solve this inequality.

Determine the set of real numbers $x$ that satisfy the following nested absolute value condition:

1. a) Solve the inequality and express the result using interval notation.

1. b) Represent the solution set on the real number line (you can use the number line in the figure, which is left blank, as a guide).

1. c) Describe the solution as a union of (closed) neighborhoods, stating the center and radius of each one.

To solve the equation $|x - 2| = |x + 6|$, three different strategies are proposed in class:

• Method A (Geometric): Interpret $|x - 2|$ as the distance from $x$ to the point $2$, and $|x + 6| = |x - (-6)|$ as the distance from $x$ to the point $-6$, deducing that $x$ must be the midpoint between both.

• Method B (Direct algebraic): Solve the equality considering that the expressions inside the absolute value must be either equal or opposites: $x - 2 = x + 6$ or $x - 2 = -(x + 6)$.

• Method C (Squaring): Since both sides are non-negative, square both sides: $(x - 2)^2 = (x + 6)^2$.

1. a) Apply the three methods independently to find the solution and verify that the same value of $x$ is reached.
2. b) Compare the three methods: which one is more direct and conceptually advantageous in this exercise? Justify your answer.
3. c) If the equation were $|x - 2| = |2x + 6|$, would Method A still be just as direct? Explain with reasons why in this new case Method B or Method C are preferable.

In an economic study on the labour market, an analyst predicts that the annual variation in the wage index $s$ (in percentage) will remain within a moderate margin if it satisfies the condition $|s - 2.5| \le 1.2$. At the same time, a macroeconomic report warns that imbalances will arise if this wage variation falls outside the stability range defined by the condition $|2s - 4| > 3$.

1. a) Solve both inequalities algebraically and independently, and express the solution of each using interval notation.
2. b) Determine the set of values of the wage increase $s$ for which both conditions are simultaneously met (that is, the analyst's forecast is met, but at the same time, imbalances are generated). Express the result as a single interval.
3. c) If collective bargaining agrees on a wage increase of $s = 3.6\,\%$, reason from the definition of absolute value as distance whether this increase satisfies the analyst's condition without leading to macroeconomic imbalances, without needing to redo the analytical calculations.