Integers on the Number Line
Badwater Basin in Death Valley is about 282 feet below sea level. Denver, the Mile High City, sits about 5,280 feet above sea level. If sea level is $0$ on a number line, how far apart are the two places? By the end of this page you will know how to answer that, and a lot more.
Plotting integers
A number line is a straight line with equally spaced tick marks. The point $0$ is the origin. Positive integers go to the right of $0$, negative integers go to the left, and the arrows on the ends say the line keeps going forever.
To read a point, look at the nearest labeled tick and count the steps. To plot a number, start at $0$ and count that many steps right (positive) or left (negative). Always check the scale first: if the labels go $0, 5, 10, \dots$, each tick may not be one unit.
Look at the following number line and identify the values represented by the letters $A$, $B$, and $C$.
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 plot the following integers on it: $$-4, \quad +3, \quad 0, \quad -1, \quad +5$$
Draw a number line in your notebook and plot the following integers on it:
$$-4, \quad +3, \quad 0, \quad -1, \quad +5$$Four points represented by letters have been marked on the following number line. Identify which integer corresponds to each of the letters $A$, $B$, $C$, and $D$.
Four points represented by letters have been marked on the following number line. Identify which integer corresponds to each of the letters $A$, $B$, $C$, and $D$.
Opposites and absolute value
Two numbers are opposites when they are the same distance from $0$ but on opposite sides. The opposite of $4$ is $-4$, and the opposite of $-4$ is $4$. We write $-(-4) = 4$. The opposite of $0$ is $0$ itself.
Distance is never negative, so an absolute value is never negative: $|a| \ge 0$ for every $a$. Opposites always have the same absolute value.
Plot the number $-3$ and its opposite on the number line. What is the distance from each to the origin?
Plot the number $-3$ and its opposite on the number line. What is the distance from each to the origin?
Plot on the number line all the integers whose absolute value is equal to $4$. How many numbers did you mark?
Plot on the number line all the integers whose absolute value is equal to $4$. How many numbers did you mark?
Given the integers $x$ such that their distance to the origin ($0$) is $6$ units: a) Plot these numbers on the number line. b) Plot their opposites as well. c) What do…
Given the integers $x$ such that their distance to the origin ($0$) is $6$ units:
- a) Plot these numbers on the number line.
- b) Plot their opposites as well.
- c) What do you observe regarding the position of the original numbers and their opposites on the number line?
Comparing and ordering integers
On a horizontal number line, numbers get greater as you move right. So if $a$ is to the left of $b$, then $a < b$. That is why $-9 < -4$: it is colder at $-9^\circ\text{F}$ than at $-4^\circ\text{F}$.
To order several numbers, first simplify each one (evaluate any opposite or absolute value), then plot them, then read from left to right.
Plot the following numbers on the number line and use it to order them from least to greatest using the $<$ sign: $$-6, \quad +2, \quad -3, \quad +4, \quad -1$$
Plot the following numbers on the number line and use it to order them from least to greatest using the $<$ sign:
$$-6, \quad +2, \quad -3, \quad +4, \quad -1$$Draw a number line and plot the following elements on it. Then, order them from least to greatest using the symbol $<$: The number $-4$. The opposite of $3$. The…
Draw a number line and plot the following elements on it. Then, order them from least to greatest using the symbol $<$:
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The number $-4$.
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The opposite of $3$.
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The absolute value of $-5$.
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The number $0$.
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The opposite of the opposite of $-2$.
Moving on the number line
Adding and subtracting integers is just moving. A positive change moves you right (or up, east, forward, above sea level). A negative change moves you left (or down, west, backward, below sea level). Each move is an arrow that starts where the last one ended.
Vertical number lines work the same way, like a thermometer: up is positive, down is negative.
A thermometer shows a temperature of $4^\circ\text{C}$ at 10:00 a.m. If by 8:00 p.m. the temperature has dropped by $6^\circ\text{C}$, plot both temperatures on a number…
A thermometer shows a temperature of $4^\circ\text{C}$ at 10:00 a.m. If by 8:00 p.m. the temperature has dropped by $6^\circ\text{C}$, plot both temperatures on a number line and indicate the final temperature.
A diver is at a depth of $8$ feet below sea level. If they decide to ascend $3$ feet to observe some fish, at what depth are they now? Represent the movement using…
A diver is at a depth of $8$ feet below sea level. If they decide to ascend $3$ feet to observe some fish, at what depth are they now? Represent the movement using arrows on a horizontal number line.
A food delivery driver starts his shift at "Point Zero" on a main avenue. He makes the following trips: 1. He travels $4$ miles to the east ($+4$). 2. From there, he…
A food delivery driver starts his shift at "Point Zero" on a main avenue. He makes the following trips:
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He travels $4$ miles to the east ($+4$).
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From there, he travels $7$ miles to the west.
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Finally, he travels another $2$ miles to the west to deliver his last order.
Draw a number line in your notebook, represent each movement using arrows, and determine at which mile marker he is when he finishes.
Distance between two points
How far is it from $-6$ to $3$? Count the steps from $-6$ to $0$ (that is $6$) and then from $0$ to $3$ (that is $3$). The distance is $6 + 3 = 9$.
There are two handy shortcuts. If the points are on opposite sides of $0$, add their absolute values. If they are on the same side, subtract the smaller absolute value from the larger.
If a point is equidistant from two others, it is their midpoint: it sits exactly halfway between them.
At a ski resort in Sierra Nevada, the thermometer recorded the following temperatures throughout the morning: At 8:00 a.m.: $-6$ $^\circ$F At 10:00 a.m.: $-2$ $^\circ$F…
At a ski resort in Sierra Nevada, the thermometer recorded the following temperatures throughout the morning:
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At 8:00 a.m.: $-6$ $^\circ$F
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At 10:00 a.m.: $-2$ $^\circ$F
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At 12:00 p.m.: $+3$ $^\circ$F
- a) Plot the three temperatures on a vertical number line (as if it were a thermometer).
- b) What was the temperature variation (distance on the line) between 8:00 a.m. and 12:00 p.m.?
Find the integer that is equidistant from $-7$ and $+3$ on the number line. Graph it to check your result and explain how you calculated it.
Find the integer that is equidistant from $-7$ and $+3$ on the number line. Graph it to check your result and explain how you calculated it.
On the integer number line, point $A$ represents the number $-4$. Point $B$ is a number such that its distance to the origin is three times the distance from $A$ to the…
On the integer number line, point $A$ represents the number $-4$. Point $B$ is a number such that its distance to the origin is three times the distance from $A$ to the origin. If point $C$ is the midpoint of the line segment formed by the opposite points of $A$ and $B$, what numbers could $C$ represent? Keep in mind that there are two possible solutions for $B$. Represent them on a graph.
Inequalities and graphing sets of integers
A condition such as $-4 \le x < 1$ describes many integers at once. To graph it, mark every integer that makes the statement true.
When $x$ can be any number, you shade a whole ray or segment. When the problem says "integers," you only mark dots on the integers, as above.
Absolute value inequalities are distance conditions. $|x| < 3$ means "less than $3$ units from $0$." $|x| \ge 2$ means "at least $2$ units from $0$." In general, $|x - a| < k$ means "less than $k$ units from $a$."
Write down in your notebook all the integers $x$ that satisfy the condition $-2 < x \leq 3$ and plot them on the number line. Pay close attention to open and closed…
Write down in your notebook all the integers $x$ that satisfy the condition $-2 < x \leq 3$ and plot them on the number line. Pay close attention to open and closed endpoints.
Write all the integers $x$ that simultaneously satisfy the following two conditions and represent them on a number line: $-5 < x \leq 2$ The absolute value of $x$ is…
Write all the integers $x$ that simultaneously satisfy the following two conditions and represent them on a number line:
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$-5 < x \leq 2$
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The absolute value of $x$ is greater than $1$, that is, $|x| > 1$.
Plot the number $-3$ on the number line and all integers whose absolute value is less than it. Is it possible to carry out this task? Justify your answer based on the…
Plot the number $-3$ on the number line and all integers whose absolute value is less than it. Is it possible to carry out this task? Justify your answer based on the definition of absolute value and its graphical representation.
Absolute value equations on the number line
The equation $|x| = 4$ has two solutions, $4$ and $-4$: the two points that are $4$ units from $0$.
The same idea works around any center. $|x - a| = k$ means "the distance from $x$ to $a$ is $k$," so $x$ is $k$ units to the right or $k$ units to the left of $a$.
If $k = 0$, there is only one solution (the center itself). If $k < 0$, there is no solution.
Given the following expressions, plot the values of $x$ that make them true on the number line: a) $|x - 3| = 5$ b) $|x + 2| = 4$ Which integer is a solution to both…
Given the following expressions, plot the values of $x$ that make them true on the number line:
- a) $|x - 3| = 5$
- b) $|x + 2| = 4$
Which integer is a solution to both expressions at the same time? Mark it with a different color on your number line.
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…
Three points $P$, $Q$, and $R$ are located on the number line. It is known that:
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Point $Q$ is the opposite of $-6$.
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Point $P$ is at a distance of $10$ units from $Q$, to the left.
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Point $R$ satisfies that $|R| = |P| + |Q|$.
Find the values of $P$, $Q$, and $R$. Represent all possible positions of $R$ on the number line, and explain which one is farthest from the midpoint between $P$ and $Q$.
Combining several conditions
Often a problem gives two or three conditions, and the answer must satisfy all of them. Think of each condition as a filter: list the candidates from the first one, then cross out the ones that fail the others. The survivors are in the overlap of the sets.
To use a phrase like "the distance between $n$ and $-2$ is greater than $5$," write it as $|n - (-2)| > 5$, or $|n + 2| > 5$. That means $n$ is more than $5$ units from $-2$, so it is either less than $-7$ or greater than $3$.
Identify all the integers $n$ that simultaneously satisfy the following three conditions and place them on a number line: 1. The absolute value of $n$ is less than or…
Identify all the integers $n$ that simultaneously satisfy the following three conditions and place them on a number line:
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The absolute value of $n$ is less than or equal to $7$: $|n| \leq 7$.
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$n$ is strictly greater than its opposite: $n > -n$.
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The distance between $n$ and the number $-2$ is greater than $5$ units.
Draw a number line and graph the following sets of numbers so that the difference between them can be seen: Set $X$: Integers $x$ such that $|x| < 4$. Set $Y$: Integers…
Draw a number line and graph the following sets of numbers so that the difference between them can be seen:
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Set $X$: Integers $x$ such that $|x| < 4$.
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Set $Y$: Integers $y$ such that their opposite $-y$ is between $-2$ and $5$ (both excluded).
Explain with reasoning which integers belong to both sets at the same time.
Solving clue puzzles
Harder problems describe positions with words. Translate each clue into a number, an expression, or a distance, then combine them.
Back to the start: with sea level as $0$, Death Valley is at $-282$ and Denver is at $5{,}280$. The distance is $|5{,}280 - (-282)| = 5{,}280 + 282 = 5{,}562$ feet.
An explorer is in a network of underground caves represented on a horizontal number line. The origin $0$ is the entrance to the cave. We know that the explorer made the…
An explorer is in a network of underground caves represented on a horizontal number line. The origin $0$ is the entrance to the cave. We know that the explorer made the following successive movements:
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He starts at an unknown point $x$.
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He moves forward $+12$ units.
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He moves backward twice his initial position $x$.
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He moves forward the opposite of $-5$.
If he ends up at point $-3$, represent the situation on the number line and determine the value of the starting point $x$, explaining your reasoning.
A submarine and a rescue helicopter are located on the same vertical line. If we plot their positions on a horizontal number line where $0$ is sea level: The submarine…
A submarine and a rescue helicopter are located on the same vertical line. If we plot their positions on a horizontal number line where $0$ is sea level:
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The submarine is at point $S$.
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The helicopter is at point $H$.
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It is known that $|S| = |H| - 200$.
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The distance between the submarine and the helicopter is $500$ feet.
Draw the number line and find the coordinates of $S$ and $H$, knowing that the helicopter is flying above sea level and the submarine is submerged.
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 twice the…
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 twice 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 $a + b$?