The absolute value of a number
is denoted by placing that number inside two vertical bars.
So for example, |3| is the absolute value of 3, which is again 3.
Similarly, |−3| is the absolute value of −3, which is also 3.
Sometimes people think that the point of the absolute value bars
is to remove minus signs, or to change minus signs to plus signs.
But that is *not* true!
For example, |2 − 6| means |−4|, which is 4,
while |2 + 6| and 2 + 6 are both 8.
This is especially important in algebra,
where you *cannot* change |2 − *x*|
to |2 + *x*| or 2 + *x*,
nor can you change |2 + *x*| to 2 + *x*.

In particular, there is no way to change |*x*| to anything simpler,
unless you happen to know whether *x* might be positive or negative.
If you do know, then you can say this:

If … | then … |
---|---|

x is positive (or zero), |
|x| = x. |

x is negative (or zero), |
|x| = −x. |

For example, in the number line below, the positive direction is to the right and the negative direction is to the left (as is usual). Accordingly, since 2 is the left of 3, 2 is less than 3, which is written 2 < 3. Similarly, since 1 is to the right of −4, 1 is greater than −4, which is written 1 > −4. Finally, since 2 is of course the same as 2, 2 is equal to 2, written 2 = 2.

Given any two expressions for real numbers, exactly one of the three symbols ‘<’, ‘>’, and ‘=’ is appropriate to put between them. If you can see the numbers on a number line, it's obvious which symbol to use. Otherwise, you can calculate it by subtracting, as in this table:

If a − b is … |
then … |
---|---|

positive, | a > b. |

negative, | a < b. |

zero, | a = b. |

The symbols ‘<’, ‘>’, and ‘=’ give complete information about the relative order of two real numbers. Sometimes you only have partial information; then you can use the symbols ‘≤’, ‘≥’, and ‘≠’, as follows:

If … | then … | or … | but not … |
---|---|---|---|

a ≤ b, | a < b, |
a = b, | a > b. |

a ≥ b, | a > b, |
a = b, | a < b. |

a ≠ b, |
a < b, | a > b, |
a = b. |

You can also compare complicated expressions built using the various operations. For example, to compare 3 + 4 and 2 · 5, you calculate that 3 + 4 is 7, while 2 · 5 is 10; since 7 < 10, you can conclude that 3 + 4 < 2 · 5. In summary:

- original problem: 3 + 4 ? 2 · 5;
- work out the left side: 3 + 4 = 7;
- work out the right side: 2 · 5 = 10;
- compare the results: 7 < 10;
- answer the original question: 3 + 4 < 2 · 5.

For a more complicated example, compare |3 − 4| with 2 · 3 − 5:

- original problem: |3 − 4| ? 2 · 3 − 5;
- work out the left side: |3 − 4| = |−1| = 1;
- work out the right side: 2 · 3 − 5 = 6 − 5 = 1;
- compare the results: 1 = 1;
- answer the original question: |3 − 4| = 2 · 3 − 5.

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