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Algebra: Inequalities with Absolute Values

Inequalities with Absolute Values

What is it about absolute values? Just when you get used to doing something one way, along come these little teeny bars that demand things be done their way. They are just like that high-maintenance boyfriend or girlfriend, who wasn't content to let you live your life the way you always had. No, suddenly it was "Why can't we eat somewhere nice for a change?" or "Is there a federal law against you brushing your teeth more than once a month?" Not that these adjustments were difficult to make, it's just that sometimes having to change little things once you get into a routine can be tricky.

Back in Graphing Linear Equations, you had to split absolute value equations into two distinct, nonabsolute value equations in order to reach a solution. Similarly, you have to break absolute value inequalities into two distinct, nonabsolute value inequalities to reach a solution. However, just to make things a little worse (if that were possible), the process is different for inequalities with less than symbols and those containing greater than symbols.

Inequalities Involving "Less Than"

If you're asked to solve an inequality problem containing absolute values, and the inequality symbol is either < or ≤, here are the steps you should follow to reach a solution:

  1. Isolate the absolute value portion on the left side of the inequality. When you do, the problem should look something like this: |x + a| < b, where a and b are real numbers. (If a is negative, the problem will look like |x - a| < b, and that's okay, too.)
  2. Create a compound inequality. Rewrite the statement |x + a| < b as -b < x + a < b. In other words, drop the absolute value symbols, place a matching inequality symbol to the left of the statement, and then write the opposite of the constant to the left of that.
  3. Solve the compound inequality. Use the procedures I showed you in the previous section to solve and/or graph the compound inequality.

Example 6: Solve the inequality |2x - 1| + 3 ≤ 6 and graph the solution.

Solution: To isolate the absolute value quantity, subtract 3 from both sides.

  • |2x - 1| ≤ 3

Drop the absolute value bars, and write the opposite of 3 to the left of the expression. Between the newly added -3 and now "bar-free" expression, place a ≤ symbol, to match the one already there.

  • -3 ≤ 2x - 1 ≤ 3

Solve the compound inequality by adding 1 to each part and then dividing everything by 2.

  • -2 ≤ 2x ≤ 4
  • -1 ≤ x ≤ 2

Graph the solution by placing solid dots at -1 and 2 and connecting them, as shown in Figure 7.5.

Graphing inequalities and absolute values

Figure 7.5The graph of |2x - 1| + 3 ≤6 is the same as the graph of the compound inequality -1 ≤ x ≤ 2.

If this problem had contained < symbols instead of ≤, the procedure would have been exactly the same, just with those less than signs throughout; of course, the graph would have contained open dots, since they coordinate so nicely with those less than signs.

You've Got Problems

4 |x - 5| < 8

Inequalities Involving "Greater Than"

Absolute value inequalities containing the symbols > or ≥ are solved much like their "less than" inequality sister problems. For one thing, you must start by isolating the absolute value quantity first, and then you must rewrite the expression without absolute value bars. However, this time you won't end up with a compound inequality, because you must rewrite the expression differently.

Once you've isolated the absolute value quantity, rewrite the statement as two separate inequalities, one that looks just like the original (just without absolute value bars), and the other with the inequality sign reversed and the opposite of the constant. Basically, the expression |x + a| > b should be rewritten as the two expressions

  • x + a > b or x + a < -b

Note the word "or" between the expressions. That doesn't mean that you only have to write one or the other (they both need to be in your solution); it means that if you plug an x into either expression and it works for just one of them, then that x value is a solution to the original inequality.

Example 7: Solve the inequality |2x + 5| - 4 > -1 and graph the solution.

Solution: Start by isolating the absolute value quantity (add 4 to both sides).

  • |2x + 5| >3

Split this into two separate inequality statements. Create the first simply by removing the bars; the second requires you to reverse the inequality sign and take the opposite of that lone constant 3. (It's good form to write the word "or" between the statements.)

  • 2x + 5 > 3 or 2x + 5 < -3
Solution graph

Figure 7.6The solution graph to |2x + 5| - 4 > -1 is the same as the graphs of x > -1 and x < - 4 put together.

Now solve each basic inequality separately.

  • 2x > -2 or 2x < -8
  • x > -1 or x < -4
You've Got Problems

Problem 7: Solve the inequality |x - 4| ≥ 2 and graph the solution.

That weird-looking two-headed monster is the answer. It says that any number greater than -1 or less than -4 will make the original inequality true. Since the solution is made up of two basic inequality statements, the graph of the solution will consist of the graphs of those statements put together. Just graph both of the inequalities on the same number line, as demonstrated in Figure 7.6.

CIG Algebra

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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