SECTION 21.1: SOLVING
MATHEMATICAL EXPRESSIONS WITH INEQUALITIES
In chapter 4 you had an introduction the **"inequality
symbols."** Go back to chapter 4 if you need to review them. Here are the inquality symbols:
<:
less than
>:
greater than
≤: less than or equal to
≥: greater than or equal to
≠: not equal to |
Consider the following 2 examples containing inequalities:
ex. 1) x > 2 ex. 2) x ≤ -3
Notice the form of the above two math expressions:
**VARIABLE
followed by an INEQUALITY SYMBOL followed by a NUMBER**
The mathematical expressions, x > 2 and x ≤ -3 contain
a variable, an inequality, and a number. Also note that the variable is
isolated on one side of the inequality and the number is isolated on the other
side of the inequality.
In example 1 above, x > 2 is shorthand for the following request: **"Find all the
values of x that are greater than 2."**
Now how many solutions are
there?
x = 3 is a solution since 3 > 2;
x = 4 is a solution since 4 > 2;
x = 5 is a solution since 5 > 2;
and on, and on, and ...
It seems there are a number of solutions. But how many? I suggest to you there
are an infinite number of solutions greater than 2 that will make x > 2
true.
So how will I represent the answer to x > 2 without simply saying "any number greater than 2 will make this inequality
true" ? Mathematicians use a "number line" to represent the solutions.
This is a VISUAL representation of ALL numbers
greater than 2.
In example 2, x ≤ -3 is shorthand for the following request: **"Find all the values of x that are less than or
equal to -3."**
Now how many solutions are there?
x = -4 is a solution since
-4 ≤ -3;
x = -5 is a solution since
-5 ≤ -3;
x = -6 is a solution since
-6 ≤ -3;
and on, and on, and ...There are an infinite number of solutions! Again, we
will use a number line to represent the infinite number of solutions since we
cannot name or list all the solutions verbally. **Why can't we name or
list all the solutions verbally?**
Notice there is a closed circle around -3 and an arrow pointing to the left.
Why a closed circle? This is to suggest that we want -3 as a solution to x ≤ -3.
And the arrow to the left suggest those numbers to the left of the closed
circle will make the inequality, x ≤ -3,
true.
**WHAT YOU NEED TO KNOW FROM THIS
SECTION** |
Any inequality in the following form
**VARIABLE INEQUALITY NUMBER **
Represents its solution on a number line. |
Example 3) State what x ≥ 12 is asking and then represent its solution on a number line.
**answer:**
is asking "Find all the values of x that are greater than OR equal to 12."
Here is the graphical solution on a number line:
SECTION 21.2: THE 2 "SIGN
RULES" OF INEQUALITIES
We need to understand some interesting features of inequalities before we
tackle solving more complicated inequalities than the ones we encountered in
section 1.
We can add, subtract, divide and multiply numbers through inequalities like we
do in equations. But some surprising results occur when we multiply and
divide both sides of an inequality.
**CONSIDER THE FOLLOWING EXAMPLES OF NUMBERS COMPARED USING INEQUALITIES**
ex. 1) 24 < 44 This
is a true inequality.
24 +12 < 44 + 12 I'll add the number 12 to both sides of 24 < 44 and see if it's still
true!
36 < 56 ** Cool, the inequality is still true!**
ex. 2) 24 -7 < 44 - 7 Now
I'll subtract 7 from both sides of 24 < 44 and see if it's still true!
17 < 37 ** Cool, The inequality is still true!**
ex. 3) 24(4) < 44(4) Now
I'll multiply both sides of 24 < 44 by positive 4 and see if it's still
true!
96 < 176 ** Cool, the inequality is still true!**
ex. 4) Now
let's divide both sides of 24 < 44 by positive 4and see if it's still true!
6 < 11 **Cool, the inequality is still true!**
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PAY CLOSE ATTENTION TO THE NEXT TWO EXAMPLES
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ex. 5) 24(-3) < 44(-3) Now I'll multiply both sides of 24 < 44 by **negative 3** and see if it's
still true!
-72 < -132 **This inequality is NOT TRUE! ** **Ohhh, ohhhh! **
ex. 6) Now
let's divide both sides of 24 < 44 by negative 4 and see if it's still
true!
-6 < -11 ** The inequality is NOT TRUE! Ohhh, ohhh...**
**CONCLUSION DRAWN FROM THE LAST**
TWO INEQUALITY
EXAMPLES |
When you multiply or divide both sides of an inequality by a negative
number the resulting inequality is always false.
**To maintain the truth of the inequality you MUST switch**
the direction of the inequality
when **multiplying or dividing**
an inequality by a negative number! |
ex. 1) Given: 24 < 44 a true inequality
24(-3) < 44(-3)
I'll multiply both sides of 24 < 44 by **negative 3**.
-72 < -132
I simplified each side of the inequality.
This inequality is NOT TRUE because I multiplied by a negative number
-72 > -132
I switched the direction of the inequality. This inequality is NOW TRUE!
ex. 2) 24 < 44 is a true inequality
I'll divide both sides of 24 < 44 by negative 4.
-6 < -11 ** **
The inequality is NOT TRUE because I divided by a negative number!
-6 > -11
I switched the direction of the inequality. This inequality is NOW TRUE!
SECTION 21.3: HOW TO SOLVE
MORE COMPLICATED INEQUALITIES
So what are we to do to maintain TRUTH when we multiply or divide both sides
of an inequality by a negative number? **When we multiply or divide an
inequality by a negative number we switch the direction of the inequality so
we can maintain TRUTH! **
**RULES FOR SOLVING INEQUALITIES** |
You solve an inequality just like an equation
**BUT**
When you multiply or divide an inequality by a negative number
switch the direction of the inequality in order to maintain truth. |
ex 1) Solve -2x + 6 > 16 and graph the solutions
Since I divided by a negative number
I changed the direction of the inequality.
**HERE IS THE GRAPHICAL SOLUTION**
**Conclusion:** All numbers less than -5 will make the original
inequality -2x + 6 > 16 true. Test and you will see that the solutions are
correct!
ex. 2) Solve -5x - 12 ≤ 8 and graph the solutions
I changed the direction of the inequality! Why?
**HERE IS THE GRAPHICAL SOLUTION**
**Conclusion:** All numbers greater than or equal to -4 will make
the original inequality -5x - 12 ≤ 8 true.
Remember we solve inequalities the same way we solve equations other than the
issue of changing the direction of the inequality when we divide or multiply
by a negative number. Here's an example using the distributive property.
ex. 3) Solve -3x - 5(2x - 4) > 7x + 60 and graph the solutions
I changed the direction of the inequality! Why?
**Here is the graph of the
solution**
**Conclusion:** All numbers less than -2 will make the original
inequality -3x - 5(2x - 4) > 7x + 60 true |