This statement … | has this solution. |
---|---|

|x| < 4; |
−4 < x < 4. |

|x| > 4; |
x < −4 or x > 4. |

|x| = 4; |
x = −4 or x = 4. |

Here are the relevant rules:

This statement … | is equivalent to this statment. |
---|---|

|a| < b; |
−b < a < b. |

|a| ≤ b; |
−b ≤ a ≤ b. |

|a| > b; |
a < −b or
a > b. |

|a| ≥ b; |
a ≤ −b or
a ≥ b. |

These pairs of statements
are equivalent for any real numbers *a* and *b*,
whether positive, negative, or zero.
This means that you can substitute
any expressions, however complicated, for *a* and *b* above,
and the equivalence will be valid.
For the examples in this class,
we'll typically use
a linear expression for *a* and a constant for *b*;
then the statement on the right
will be one that you already know how to solve.

For example, here's an inequality where the absolute value of a linear expression is less than a constant:

- |
*t*− 5| < 12 — original statement; - −12 <
*t*− 5 < 12 — freed of absolute values; - −7 <
*t*< 17 — add 5 to all sides.

And here's one where the absolute value of a linear expression is greater than a constant:

- |2
*y*+ 3| > 6 — original statement; - 2
*y*+ 3 < −6 or 2*y*+ 3 > 6 — freed of absolute values; - 2
*y*< −9 or 2*y*> 3 — subtract 3 from both sides of each inequality; *y*< −9/2 or*y*> 3/2 — divide both sides of each inequality by 2.

Here's an example with a weak inequality; notice that it uses the same rule as for a strict inequality:

- |7 −
*x*| ≤ 2 — original statement; - −2 ≤ 7 −
*x*≤ 2 — freed of absolute values; - −9 ≤ −
*x*≤ −5 — subtract 7 from all sides; - 9 ≥
*x*≥ 5 — take the opposite of all sides and reverse the inequalities; - 5 ≤
*x*≤ 9 — swap the order to keep things increasing.

There are also some degenerate problems along this line. Here's an example:

- |
*n*+ 3| < −4 — original statement; - 4 <
*n*+ 3 < −4 — freed of absolute values; - 1 <
*n*< −7 — subtract 3 from all sides; - False — since 1 > −7.

This statement … | is equivalent to this statment. |
---|---|

|a| = b; |
a = −b or a = b,
and b ≥ 0. |

Here's an example to show what I mean:

- |2
*r*+ 5| = 7 — original statement; - 2
*r*+ 5 = −7 or 2*r*+ 5 = 7, and 7 ≥ 0 — freed of absolute values; - 2
*r*+ 5 = −7 or 2*r*+ 5 = 7 — since in fact 7 > 0; - 2
*r*= −12 or 2*r*= 2 — subtract 5 from both sides of each equation; *r*= −6 or*r*= 1 — divide both sides of each equation by 2.

Normally, you wouldn't even bother to write down the bit about 7 ≥ 0; since you can see right away that this is true, you go on directly to the next step. Here's an example where I do just that:

- |4
*c*− 8| = 6 — original statement; - 4
*c*− 8 = −6 or 4*c*− 8 = 6 — freed of absolute values, since 6 ≥ 0; - 4
*c*= 2 or 4*c*= 14 — add 8 to both sides of each equation; *c*= 1/2 or*c*= 7/2 — divide both sides of each equation by 4.

Still, you *do* have to think about that bit;
compare this example:

- |2
*x*− 3| = −5 — original statement; - 2
*x*− 3 = 5 or 2*x*− 3 = −5, and −5 ≥ 0 — freed of absolute values; - False — since in fact −5 < 0.

To do this, pretend that the absolute value is itself a single thing
(don't pay any attention for now to what's inside it),
and *this* thing is the variable that you're solving for.

For example:

- |
*y*+ 2| − 5 = 7 — original problem; - |
*y*+ 2| = 12 — add 5 to both sides to isolate the absolute value; *y*+ 2 = −12 or*y*+ 2 = 12 — now freed of absolute values, since 12 ≥ 0;*y*= −14 or*y*= 10 — subtract 2 from both sides of each equation.

Here's another example:

- 3|
*k*− 9| ≤ 12 — original problem; - |
*k*− 9| ≤ 4 — divide both sides by 3 to isolate the absolute value; - −4 ≤
*k*− 9 ≤ 4 — now freed of absolute values; - 5 ≤
*k*≤ 13 — add 9 to all sides.

However, if you simply want to say that the two absolute values are equal, then there is a short cut that makes everything much easier. The reason is that two real numbers have the same absolute value exactly when they are either equal or opposite:

This statement … | is equivalent to this statment. |
---|---|

|a| = |b|; |
a = b or
a + b = 0. |

For example:

- |2
*x*+ 4| = |*x*− 9| — original problem; - 2
*x*+ 4 =*x*− 9 or (2*x*+ 4) + (*x*− 9) = 0 — freed of absolute values; *x*+ 4 = −9 or 3*x*− 5 = 0 — subtract*x*from both sides of the first equation, and simplify the left side of the second equation;*x*= −13 or 3*x*= 5 — subtract 4 from both sides of the first equation, and add 5 to both sides of the second equation;*x*= −13 or*x*= 5/3 — divide both sides of the second equation by 3.

Occasionally you can get a degenerate one of these too. For example:

- |3
*t*+ 4| = |3*t*+ 2| — original problem; - 3
*t*+ 4 = 3*t*+ 2 or (3*t*+ 4) + (3*t*+ 2) = 0 — freed of absolute values; - 4 = 2 or 6
*t*+ 6 = 0 — subtract 3*t*from both sides of the first equation, and simplify the left side of the second equation; - 6
*t*+ 6 = 0 — in fact 4 > 2; - 6
*t*= −6 — subtract 6 from both sides of the remaining equation; *t*= −1 — divide both sides of the equation by 6.

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