Exercises: Representation on the number line

Look at the following number line and identify the values represented by the letters $A$, $B$ and $C$.

Draw a number line in your notebook and represent the following integers on it:

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

Represent the number $-3$ and its opposite on the number line. How far is each one from the origin?

You can use this number line, in which the origin and the units are marked (each mark is one unit), or draw your own:

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

A thermometer shows a temperature of $4^\circ\text{C}$ at ten in the morning. If at eight in the evening the temperature has dropped by $6^\circ\text{C}$, represent both temperatures on a line and indicate the final temperature.

Write in your notebook all the integers $x$ that satisfy the condition $-2 < x \leq 3$ and represent them on the number line. Pay close attention to the open and closed endpoints.

You can copy this number line into your notebook:

Represent on the number line all the integers whose absolute value is equal to $4$. How many numbers have you marked?

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

A scuba diver is at a depth of $8$ metres below sea level. If they decide to go up $3$ metres to look at some fish, what depth are they at now? Represent the movement using arrows on a horizontal number line.

Place the following numbers on the number line and use it to order them from least to greatest using the symbol $<$:

During a cold snap recorded by AEMET, the minimum temperature in Teruel was $-6$ ºC and the maximum in Valencia was $9$ ºC. Lucas wants to calculate the temperature difference (the distance between both values on the number line) and writes in his notebook:

Lucas concludes that the distance between both temperatures on the number line is $3$ units.

1. a) Find and explain in your own words the conceptual error Lucas made when translating the distance between two numbers into an absolute value operation.
2. b) Write the correct expression using absolute value and calculate with reasoning the real distance between both temperatures.
3. c) Show that subtracting the temperatures in reverse order inside the absolute value yields exactly the same result and justify why this happens.

Two divers from a marine biology team in Cabo de Gata communicate their vertical position relative to sea level (which we take as elevation $0$ m): Marcos is submerged at elevation $-24$ m and Lucía is on a platform above the surface at elevation $+7$ m. The goal is to determine the vertical distance $|a - b|$ between them.

1. a) Without doing the formal subtraction yet, estimate between which two consecutive tens (for example, between $10$ and $20$, between $20$ and $30$, etc.) this distance must lie, and explain why it is impossible for the result to be less than $24$ m.
2. b) Apply the formal formula for the distance between two integers to calculate the exact distance between Marcos and Lucía.
3. c) Check whether the exact result matches your estimate and justify why a distance on the number line can never be a negative number.

On the following number line, four points represented by letters have been marked. Identify which integer corresponds to each of the letters $A$, $B$, $C$, and $D$.

Write down all the integers $x$ that satisfy the following two conditions simultaneously and represent them on a number line:

• $-5 < x \leq 2$

• The absolute value of $x$ is greater than $1$, that is, $|x| > 1$.

A food delivery driver starts their workday at the "Zero Point" of a large main avenue. They make the following movements:

1. They travel $4$ kilometers east ($+4$).

2. From there, they move $7$ kilometers west.

3. Finally, they go another $2$ kilometers west to deliver their last order.

Draw a number line in your notebook, represent each movement with arrows, and determine at which kilometer point they are at the end.

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

Represent the number $-3$ and all the integers whose absolute value is less than it on the number line. Is it possible to complete this task? Justify your answer based on the definition of absolute value and its graphical representation.

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

Given the integers $x$ whose distance from the origin ($0$) is $6$ units, and using the number line in the figure (or drawing an identical one in your notebook):

1. a) Represent those numbers on the number line.

1. b) Also represent their opposites.

1. c) What do you notice about the position of the original numbers and their opposites on the line?

In a ski resort in Sierra Nevada, the thermometer recorded the following temperatures throughout the morning:

• At 8:00: $-6$ $^\circ$C

• At 10:00: $-2$ $^\circ$C

• At 12:00: $+3$ $^\circ$C

1. a) Represent the three temperatures on a vertical number line (like a thermometer).
2. b) What was the temperature change (distance on the line) between 8:00 and 12:00?

Draw a number line and place the following elements on it. Then, order them from least to greatest using the symbol $<$:

• The number $-4$.

• The opposite of $3$.

• The absolute value of $-5$.

• The number $0$.

• The opposite of the opposite of $-2$.

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

Find the integer that is exactly the same distance from $-7$ and from $+3$ on the number line. In the figure those two numbers are marked. Represent it graphically on the line to check your result and explain how you calculated it.

In a maritime shipping channel, two acoustic warning buoys have been placed at integer positions $-8$ and $+6$ on the number line. A repeater station needs to be anchored at an intermediate integer point $c$ such that both buoys are at exactly the same distance $d$ from this repeater.

1. a) Deduce with reasoning the central position $c$ of the repeater and calculate the value of the distance $d$ to either buoy.
2. b) Write an absolute value equation of the form $|x - c| = d$ whose only solutions on the number line accurately represent the positions of both buoys.
3. c) Solve your equation algebraically, verifying that it gives exactly the points $-8$ and $+6$, and explain why an equation of this type always yields two symmetric solutions with respect to $c$.

An AEMET weather station located in Sierra Nevada has an automatic frost alarm programmed. The sensor activates the warning signal if the integer temperature $T$ (in degrees Celsius) falls within the risk range from $-7$ ºC to $+2$ ºC, with the condition that the boundary value $-7$ ºC does activate the alert, whereas if the temperature rises to reach exactly $+2$ ºC, the alert is deactivated as it is considered out of danger.

1. a) Write the activation condition as a double inequality for the variable $T$ (an interval).
2. b) Explain what type of geometric representation (open circle or solid circle) corresponds to each of the two endpoints on the number line and what physical meaning each one has in this context.
3. c) Write the complete list of all integer temperature values that activate the alarm and calculate the distance on the number line between both endpoints using subtraction within an absolute value.

An explorer is in a network of underground caves represented on a horizontal number line. The origin $0$ is the cave entrance. We know that the explorer has made the following successive movements:

1. Starts at an unknown point $x$.

2. Moves forward $+12$ units.

3. Moves back twice their initial position $x$ (that is, $2x$ units to the left).

4. Moves forward the opposite of $-5$.

If they finally end up at point $-3$, represent the situation on the number line (draw the line and the movements yourself) and determine, with reasoning, the value of the starting point $x$.

On the number line of integers, point $A$ represents the number $-4$. Point $B$ is a number such that its distance from the origin is three times the distance of $A$ from the origin. If point $C$ is the midpoint of the line segment formed by the opposite points of $A$ and $B$, which numbers could $C$ represent? Note that there are two possible solutions for $B$. Represent them graphically.

Identify all the integers $n$ that satisfy the following three conditions simultaneously and plot them on a number line:

1. The absolute value of $n$ is less than or equal to $7$: $|n| \leq 7$.

2. $n$ is strictly greater than its opposite: $n > -n$.

3. The distance between $n$ and the number $-2$ is greater than $5$ units.

A submarine and a rescue helicopter are positioned on the same vertical line. If we transpose their positions to a horizontal number line where $0$ is sea level:

• The submarine is located at point $S$.

• The helicopter is located at point $H$.

• It is known that $|S| = |H| - 200$.

• The distance between the submarine and the helicopter is $500$ metres.

Draw the number line and find the coordinates of $S$ and $H$, knowing that the helicopter flies above sea level and the submarine is submerged.

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

Three points $P$, $Q$, and $R$ are located on the number line. It is known that:

• Point $Q$ is the opposite of $-6$.

• Point $P$ is at a distance of $10$ units from $Q$, to the left.

• Point $R$ satisfies $|R| = |P| + |Q|$.

Calculate the values of $P$, $Q$, and $R$. Represent all possible positions of $R$ on the line and explain which of them is furthest from the midpoint between $P$ and $Q$.

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

Given the following expressions, place on the number line the values of $x$ that make them true:

1. a) $|x - 3| = 5$
2. b) $|x - 2| = 4$

Which integer is a solution to both expressions at the same time? Mark it with a different colour on your line.

Consider a negative integer $a$ and a positive integer $b$. On the number line, the distance between $a$ and $b$ is $15$ units. If the absolute value of $a$ is double the absolute value of $b$, that is, $|a| = 2 \cdot |b|$, find the values of $a$ and $b$ and represent them. What is the opposite of the sum $a + b$?

Draw a number line and represent the following sets of numbers so that the difference between them can be seen (for example, by using a different color for each set):

• Set $X$: integers $x$ such that $|x| < 4$.

• Set $Y$: integers $y$ such that their opposite $-y$ is between $-2$ and $5$ (both excluded).

State, with reasoning, which integers belong to both sets at the same time.

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

The goal is to determine all integer numbers $x$ whose distance to the number $-3$ on the number line is less than or equal to $5$ units. To solve it, two students propose different strategies:

• Irene's method (geometric): She places $-3$ on the number line, counts $5$ units to the right and $5$ to the left, draws the endpoints with solid circles, and writes down all the integers that lie within that section.

• David's method (algebraic): He sets up the absolute value inequality $|x - (-3)| \le 5$, simplifies it to $|x + 3| \le 5$, splits it into the double inequality $-5 \le x + 3 \le 5$, and solves for the unknown $x$ by subtracting $3$ from each part.

1. a) Solve the problem using both methods and verify that you reach exactly the same set of integers.
2. b) Suppose now that the problem asks to find the integers whose distance to the number $-430$ is less than or equal to $125$ units. Compare both methods and justify which of the two is preferable in this second case, pointing out the risks and drawbacks of the other method.

On an automated metro line, the different stops are represented as integer points on a number line. A repeater antenna is to be installed at an integer position $k$. The antenna's coverage reaches any point $x$ whose distance to $k$ is at most $7$ units; that is, the points that satisfy $|x - k| \le 7$. The technical management establishes two conditions:

1. It must cover all stations in the critical tunnel section, which includes all integers in the interval $[-11, -5]$ (both endpoints included).

2. Under no circumstances must it reach the central transfer station, located at point $0$ (the origin), to avoid interfering with trackside computers.

1. a) From the first condition, analyze what distance there must be from $k$ to the endpoint $-11$ and to the endpoint $-5$, and determine what integer values the position $k$ can take.
2. b) Analyze the second condition to determine which positions of $k$ would cause point $0$ to be within the coverage radius, and discard them.
3. c) Conclude what the only valid integer locations are to install the antenna and, for one of them, write the complete interval of stations that will be covered.