Beyond Equations: Inequalities & Absolute Value
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In This Chapter…
An Inequality With a Variable: A Range on the Number Line
Many Values “Solve” an Inequality
Solve Inequalities: Isolate Variable by Transforming Each Side
Multiply an Inequality by a Negative: Flip > to < or Vice Versa
Absolute Value: The Distance from Zero
Replace |x| with x in One Equation and with-x in Another
Inequalities + Absolute Values: Set Up Two Inequalities
Chapter 8:
Beyond Equations:
Inequalities & Absolute Value
In This Chapter:
• Introduction to inequalities
• Solving inequalities
• Working with absolute value
An Inequality With a Variable: A Range on the Number Line
Inequalities use <, >, ≤ or ≥ to describe the relationship between two expressions.
y≤7
5>4
x<5
2x + 3 ≥ 0
Like equations, inequalities are full sentences. Always read from left to right.
x
x>y
x≤y
x≥y
x is less than y
x is greater than y
x is less than or equal to y
x is greater than or equal to y
x is at most y
x is at least y
We can also have two inequalities in one statement. Just make a compound sentence.
9 < g < 200
–3 < y ≤ 5
7≥x>2
9 is less than g, and g is less than 200
–3 is less than y, and y is less than or equal to 5
7 is greater than or equal to x, and x is greater than 2
To visualize an inequality that involves a variable, represent the inequality on a number line.
Recall that “greater than” means “to the right of” on a number line. Likewise, “less than” means “to
the left of.”
y is to the right of 5. 5 is not included in the
line (as shown by the empty circle around 5),
because 5 is not a part of the solution—y is
greater than 5, but not equal to 5.
b is to the left of 2 (or on top of 2). Here,
2 is included in the solution, because b
can equal 2. A solid black circle indicates
that you include the point itself.
Any number covered by the black arrow will make the inequality true and so is a possible solution to
the inequality. Any number not covered by the black arrow is not a solution.
If you…
Want to visualize an
inequality
Then you…
Put it all on a number line, where < means “to the left of”
and > means “to the right of”
Like this:
Check Your Skills
Represent the following equations on the number line provided:
1. x > 3
2. b ≥ –2
3. y = 4
Translate the following into inequality statements:
4. z is greater than v.
5. The total amount is greater than $2,000.
Answers can be found on page 351.
Many Values “Solve” an Inequality
What does it mean to “solve an inequality”?
The same thing as it means for an equation: find the value or values of x that make the inequality true.
When you plug a solution back into the original equation or inequality, you get a true statement.
Here's what's different. Equations have only one (or just a few) values as solutions. In contrast,
inequalities give a whole range of values as solutions—way too many to list individually.
Equation: x + 3 = 8
Inequality: x + 3 < 8
The solution to x +
3 = 8 is x = 5. 5 is The solution to x + 3 < 8 is x < 5. Now, 5 itself is not a solution
the only number that because 5 + 3 < 8 is not a true statement. But 4 is a solution because 4
+ 3 < 8 is true. For that matter, 4.99, 3, 2, 2.87, –5, and –100 are also
will make
the equation true. solutions. The list goes on. Whichever of the correct answers you
plug in, you arrive at something that looks like this:
Plug back in to
(Any number less than 5) + 3 < 8. True.
check:
5 + 3 = 8. True.
Check Your Skills
6. Which of the following numbers are solutions to the inequality x < 10?
(A) –3
(B) 2.5
(C) –3/2
(D) 9.999
(E) All of the above
The answer can be found on page 351.
Solve Inequalities: Isolate Variable by Transforming Each Side
As with equations, your objective is to isolate our variable on one side of the inequality. When the
variable is by itself, you can see what the solution (or range of solutions) really is.
2x + 6 < 12 and x < 3 provide the same information. But you understand the full range of solutions
more easily when you see the second inequality, which literally says that “x is less than 3.”
Many manipulations are the same for inequalities as for equations. First of all, you are always
allowed to simplify an expression on just one side of an inequality. Just don't change the expression's
value.
2x + 3x < 45
is the same as
5x < 45
The inequality sign isn't involved in this simplification.
Next, some “Golden Rule” moves work the same way for inequalities as for equations. For instance,
you can add anything you want to both sides of an inequality. Just do it to both sides. You can also
subtract anything you want from both sides of an inequality.
We can also add or subtract variables from both sides of an inequality. It doesn't matter what the sign
of the variables is.
If you…
Then you…
Want to add or subtract something on both sides of an
inequality
Go ahead and
do it
Check Your Skills
Isolate the variable in the following inequalities.
Like this:
7. x – 6 < 13
8. y + 11 ≥ –13
9. x + 7 > 7
Answers can be found on page 351.
Multiply an Inequality by a Negative: Flip > to < or Vice Versa
If you multiply both sides of an inequality by a positive number, leave the inequality sign alone. The
same is true for division.
However, if you multiply both sides of an inequality by a negative number, flip the inequality
sign. “Greater than” becomes “less than” and vice versa.
–2x > 10
–b ≥ 10
–1 × (–b) ≤ (10) × (–1)
b ≤ –10
x < –5
If you didn't switch the sign, then true inequalities such as 5 < 7 would become false when you
multiplied them by, say, –1. You must flip the sign.
5<7
5 is less than 7
but
–5 > –7
–5 is greater than –7
What about multiplying or dividing an inequality by a variable? The short answer is…try not to do it!
The issue is that you don't know the sign of the “hidden number” that the variable represents. If the
variable has to be positive (e.g., it counts people or measures a length), then you can go ahead and
multiply or divide. But still keep your eyes peeled for danger.
If you…
Then you…
Multiply or divide both sides of an
inequality
by a number
Flip the inequality sign if the
number is
negative
Check Your Skills
Isolate the variable in each equation.
Like
this:
10. x + 3 ≥ –2
11. –2y < –8
12. a + 4 ≥ 2a
Answers can be found on page 351.
Absolute Value: The Distance from Zero
The absolute value of a number describes how far that number is away from 0. It is the distance
between that number and 0 on a number line.
The symbol for absolute value is | number |. For instance, write the absolute value of –5 as |–5|.
The absolute value of 5 is 5.
|5| = 5
The absolute value of –5 is also 5.
|–5| = 5
When you face an expression like |4 – 7|, treat the absolute value symbol like parentheses. Solve the
arithmetic problem inside first, and then find the absolute value of the answer.
|4 – 7| = ?
|–3| = 3
4 – 7 = –3
Check Your Skills
Mark the following expressions as TRUE or FALSE.
13. |3| = 3
14. |–3| = –3
15. |3| = –3
16. |–3| = 3
17. |3 – 6| = 3
18. |6 – 3|= –3
Answers can be found onpages 351–352.
Replace |x| with x in One Equation and with–x in Another
You often see a variable inside the absolute value signs.
Example: |y| = 3
This equation has two solutions. There are two numbers that are 3 units away from 0: namely, 3 and –
3. So both of these numbers could be possible values for y. y is either 3 or –3.
When you see a variable inside an absolute value, look for the variable to have two possible values.
Here is a step–by–step process for finding both solutions.
Step 1: Isolate the absolute value expression on one side of the equation. Here,
|y| = 3
the expression is already isolated.
Step 2: Drop the absolute value signs and set up two equations. The first
+
equation has the positive value of what's inside the absolute value. The second
(y) = 3 or – (y) = 3
equation puts in a negative sign.
y = 3 or –y = 3
Step 3: Solve both equations.
y = 3 or y = –3
y = 3 or y = –3
You have two possible values.
You can take a shortcut and go right to “ y equals plus or minus 3.” This shortcut works as long as the
absolute value expression is by itself on one side of the equation.
Here's a slightly more difficult problem:
Example: 6 × |2x + 4| = 30
To solve this problem, you can use the same approach.
6 × |2x + 4| = 30
Step 1: Isolate the absolute value expression on one side of the equation or
|2x + 4| = 5
inequality.
2x + 4 = 5 or –
Step 2: Set up two equations—the positive and the negative values are set
(2x + 4) = 5
equal to the other side.
–2x – 4 = 5
2x = 1 or –2x = 9
Step 3: Solve both equations/inequalities.
or
Note: We have two possible values.
If you…
Have a variable inside
absolute value signs
Then you…
Like this
|z| = 4
+(z) = 4 o
Drop the absolute value and set up two equations, one
(z) = 4
positive and one negative
z = 4 or z
4
Check Your Skills
Solve the following equations with absolute values in them:
19. |a| = 6
20. |x + 2| = 5
21. |3y – 4|= 17
22. 4|x + 1/2| = 18
Answers can be found on page 352.
Inequalities + Absolute Values: Set Up Two Inequalities
Some tough problems include both inequalities and absolute values. To solve these problems,
combine what you have learned about inequalities with what you have learned about absolute values.
|x| ≥ 4
The basic process for dealing with absolute values is the same for inequalities as it is for equations.
The absolute value is already isolated on one side, so now drop the absolute value signs and set up
two inequalities. The first inequality has the positive value of what was inside the absolute value
signs, while the second inequality has the negative value.
+ (x) ≥ 4 or – (x) ≥ 4
Now isolate the variable in each inequality, as necessary.
+ (x) ≥ 4
x≥4
– (x) ≥ 4
–x ≥ 4
x ≤ –4
Divide by –1
Remember to flip the sign when dividing by a negative
So the two solutions to the original equation are x ≥ 4 and x ≤ –4. Represent that on a number line.
As before, any number that is covered by the black arrow will make the inequality true. Because of
the absolute value, there are now two arrows instead of one, but nothing else has changed. Any
number to the left of –4 will make the inequality true, as will any number to the right of 4.
Looking back at the inequality |x| ≥ 4, you can also interpret it in terms of distance. |x| ≥ 4 means “x is
at least 4 units away from zero, in either direction.” The black arrows indicate all numbers for which
that statement is true.
Here is a harder example:
|y + 3| < 5
Once again, the absolute value is already isolated on one side, so set up the two inequalities.
+ (y + 3) < 5 and
– (y + 3) < 5
Next, isolate the variable.
–y – 3 < 5
–y < 8
y > –8
y+3<5
y<2
The two inequalities are y < 2 and y > –8. If you plot those results, something curious happens.
It seems as if every number should be a solution to the equation. But if you start testing numbers, you
see that 5 doesn't work, because |5 + 3| is not less than 5. In fact, the only numbers that make the
inequality true are those that are true for both inequalities. The number line should look like this:
In the first example, x could be greater than or equal to 4 OR less than or equal to –4. For this
example, however, it makes more sense to say that y is greater than –8 AND less than 2. If your two
arrows do not overlap, as in the first example, any number that falls in the range of either arrow will
be a solution to the inequality. If your two arrows overlap, as in the second example, only the
numbers that fall in the range of both arrows will be solutions to the inequality.
You can interpret |y + 3| < 5 in terms of distance: “(y + 3) is less than 5 units away from from zero, in
either direction.” The shaded segment indicates all numbers y for which this is true. As inequalities
get more complicated, don't worry about interpreting their meaning—just solve them algebraically!
If you…
Then you…
Have an inequality with variable
inside absolute value signs
Drop the absolute value and set up two
inequalities, one positive and one negative
Like th
|z| > 4
+(z) > 4
– (z) >
z > 4 or
–4
Check Your Skills
23. |x + 1| > 2
24. |–x – 4| ≥ 8
25. |x – 7| < 9
Answers can be found on pages 352–353.
Check Your Skills Answer Key:
1.
2.
3.
4. z > v
5. Let a = amount.
a > $2,000
6. The answer is (E) All of the above! All of these numbers are to the left of 10 on the number line.
7. x – 6 < 13
x < 19
8. y + 11 ≥ –13
y ≥ –24
9. x + 7 > 7
x>0
10. x + 3 ≥ –2
x ≥ –5
11. –2y < –8
y>4
12. a + 4 ≥ 2a
4≥a
13. True
14. False—(Note that absolute value is always positive!)
15. False
16. True