Please enable scripting (or JavaScript) in your web browser, and then reload this page.
In this lesson you will learn about absolute values of numbers, and study graphs of formulas involving absolute value. You will also learn how to solve equations involving absolute value.
As you know, every real number except zero has an opposite, and two numbers that are opposites of each other are the same distance from zero. For example, $-3$ and 3 are opposites, and each is 3 units away from zero.
The absolute value of a number is the distance between that number and zero. For example, the absolute value of 3 is 3 and the absolute value of $-3$ is 3. The absolute value of 3 is written $|3|$, so ${|3|}=3$ and ${|-3|}=3$.
Click to see the graph of $y={|x|}$.
Look at the graphs of $y=x$ and $y=-x$ shown below. Compare the graph of $y={|x|}$ shown on the grid to the left to these graphs.
Look at the graph of $y={|x|}$. On this graph, there are two points that have a $y$-value of 2: the points $(2,2)$ and $(-2,2)$. This means that if ${|x|}=2$, then $x$ can be either 2 or $-2$.
Look at the graph of $y={|x|}$, and answer the following questions about its shape and properties.
You have seen that if ${|x|}=3$, $x$ can be either 3 or $-3$ because both 3 and $-3$ are 3 units away from zero. In this section you will learn how to solve equations such as ${|x-2|}=3$. First, let’s see what the graph of $y={|x-2|}$ looks like.
Click to look at the graph of $y={|x-2|}$.
On the graph of $y={|x-2|}$ there are two points that have a $y$-value of 3: the points $(-1,3)$ and $(5,3)$. This means that if ${|x-2|}=3$, then $x$ can be either $-1$ or 5. Use the graph to answer the following questions.
The graph of $y={|x-2|}$ is shown in green and the graph of $w={|x|}$ is shown in red.
You can also solve equations involving absolute value algebraically. You know that if ${|x|}=3$, then $x=3$ or $x=-3$. Similarly, if ${|x-2|}=3$, then $x-2$ must equal 3 or $-3$.
Always check your solutions by substituting them back in the original equation:
Test $x=5$: ${|5-2|}={|3|}=3$. True.
Test $x=-1$: ${|-1-2|}={|-3|}=3$. True.
Solve each equation in the table below algebraically, as shown above. Check your solutions by substituting them in the original equation.
The equations $y=-\,{|x|}$ (in blue) and $w={|x|}$ (in red) are graphed on the grid to the left.
Compute ${|a|}\,{|b|}$ and $|ab|$ for the values of $a$ and $b$ in the table below.
The equations $y={|a|}\,{|x|}$ and $y={|ax|}$ are graphed on the grids to the left, with a slider for $a$.
As you can see:
For any two numbers $a$ and $b$, ${|a|}\,{|b|}={|ab|}$.
In the last question, you saw that ${|a|}\,{|b|}={|ab|}$. That is, it doesn’t matter whether you multiply two numbers first or take their absolute value first; the end result is the same either way. In this question, we’ll look at sums of absolute values.
Find two numbers $a$ and $b$ so that $|a|+{|b|}$ is not equal to $|a+b|$. (Try mixing positive and negative numbers.)
The equations $y={|x|}+{|a|}$ and $y={|x+a|}$ are graphed on the grids to the left, with a slider for $a$.