Absolute Value

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 city on the Spanish plateau 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) $|x + 2| = 0$

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

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.

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

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.

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 $3$.

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.

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.

Express using an absolute value inequality the set of real numbers represented in the following graph:

In a statistical study on the monthly leisure expenditure of young people in a city, it has been determined that the mean is $900$ pesos with a maximum error of $240$ pesos.

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 $640$ pesos per month be included in this study group? Justify your answer.

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.

Given the following inequalities, identify which one defines a neighborhood with center $a = -2$ and radius $r = 3$ and solve the resulting inequality:

1. a) $|x - 2| < 3$
2. b) $|x + 2| < 3$
3. c) $|x - 3| < 2$

Once identified, represent the complementary set graphically on the number line.

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

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.

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.

Solve the following nested absolute value equation and represent the solutions on the number line:

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 pesos. 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.

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.