and solve each for x by subtracting 5 and dividing by 4: If 4x + 5 = 13, then 4x = 8 and x = 2.
If
, then
and
.
You have two solutions: 2 and
. Both solutions work when you replace the x in the original equation with their values.
One restriction you should be aware of when applying the rule for changing from absolute value to individual linear equations is that the absolute value term has to be alone on one side of the equation.
For instance, to solve
, you have to subtract 7 from each side of the equation and then divide each side by 3. Subtracting 7, you have
; then, when you divide by 3, the problem becomes
.
Now you can write the two linear equations and solve them for x:
If 4 – 3x = 6, then –3x = 2 and
.
If 4 – 3x = –6, then –3x = –10 and
.
Seeing through absolute value inequality
An absolute value inequality contains both an absolute value,
, and an inequality: <, >, < , or >. But, then, you knew that was coming.
To solve an absolute value inequality, you have to change from absolute value inequality form to just plain inequality form. The way to handle the change from absolute value notation to inequality notation depends on which direction the inequality points with respect to the absolute value term. The methods, depending on the direction, are quite different:
✔ To solve for x in
you solve –c < ax + b < c.
✔ To solve for x in
|, you solve both ax + b > c and ax + b < –c.
The first change sandwiches the ax + b between c and its opposite. The second change considers values greater than c (toward positive infinity) and smaller than –c (toward negative infinity).
Sandwiching the values in inequalities
You apply the first rule of solving absolute value inequalities to the inequality
, because of the less-than direction of the inequality. You rewrite the inequality, using the rule for changing the format:
. Next, you add one to each section to isolate the variable; you get the inequality
. Divide each section by two to get
. You can write the solution in interval notation as [–2, 3].
Be sure that the absolute value inequality is in the correct format before you apply the rule. The absolute value portion should be alone on its side of the inequality sign. If you have
, for example, you need to add 7 to each side and divide each side by 2 before changing the form:
Adding 7 you get
, and then dividing by 2 the inequality becomes
. Applying the first rule the problem becomes –9 < 3x + 5 < 9. Subtracting 5 from each interval gives you –14 < 3x < 4. Then, dividing each interval by 3 you have
. In interval notation, the answer is written
.
Harnessing inequalities moving in opposite directions
An absolute value inequality with a greater-than sign, such as
> 11, has solutions that go infinitely high to the right and infinitely low to the left on the number line. To solve for the values that work, you rewrite the absolute value, using the rule for greater-than inequalities; you get two completely separate inequalities to solve. The solutions relate to the inequality 7 – 2x > 11 or to the inequality 7 – 2x < –11. Notice that when the sign of the value 11 changes from positive to negative, the inequality symbol switches direction.
When solving the two inequalities, be sure to remember to switch the sign when you divide by –2:
Solving 7 – 2x > 11, you first subtract 7 from each side to get –2x > 4. Dividing by –2, you have x < –2.
Solving 7 – 2x < –11, you subtract 7 to get –2x < –18. Dividing by –2, you have x > 9. The solution of the absolute value inequality is x < –2 or x > 9. In interval notation, you write the solution as
or
.
Don’t ever write the solution x < –2 or x > 9 as 9 < x < –2. If you do, you indicate that some numbers can be bigger than 9 and smaller than –2 at the same time. It just isn’t so.
Exposing an impossible inequality imposter
The rules for solving absolute value inequalities are relatively straightforward. You change the format of the inequality and solve for the values of the variable that work in the problem. Sometimes, however, amid the flurry of following the rules, an impossible situation works its way in to try to catch you off guard.
For example, say you have to solve the absolute value inequality
. It doesn’t look like such a big deal; you just subtract 8 from each side and then divide each side by 2. The dividing value is positive, so you don’t reverse the sense. After performing the initial steps, you use the rule where you change from an absolute value inequality to an inequality with the variable term sandwiched between inequalities. So, what’s wrong with that? Here are the steps:
Subtracting 8, you get
, and dividing by 2, you get
.
Under the format –c < ax + b < c, the inequality looks curious. Do you sandwich the variable term between –1 and 1 or between 1 and –1 (the first number on the left, and the second number on the right)? It turns out that neither works. First of all, you can throw out the option of writing 1 < 3x – 7 < –1. Nothing is bigger than 1 and smaller than –1 at the same time. The other version
and solve each for x by subtracting 5 and dividing by 4: If 4x + 5 = 13, then 4x = 8 and x = 2.
