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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 21: SOLVING INEQUALITIES

 

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.


Figure
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?


1

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:

$x\geq 12$ 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:

Figure

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) MATH 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) MATH   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

MATH  
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

$-2x+6>16$

MATH

$-2x>10$

MATH Since I divided by a negative number

I changed the direction of the inequality.

$\rightarrow x<-5$


HERE IS THE GRAPHICAL SOLUTION
2

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

$-5x-12\leq 8$

MATH

MATH

MATH I changed the direction of the inequality! Why?

MATH


HERE IS THE GRAPHICAL SOLUTION
3

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

$-3x-5(2x-4)>7x+60$

MATH

MATH

MATH

MATH

MATH

MATH

MATH I changed the direction of the inequality! Why?

$\rightarrow x<-2$

Here is the graph of the solution
4

Conclusion: All numbers less than -2 will make the original inequality -3x - 5(2x - 4) > 7x + 60 true

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE

- CHAPTER 1: THE NUMBER SYSTEM

- CHAPTER 2: ORDER OF OPERATIONS

- CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS

- CHAPTER 4: SIGNED NUMBERS, ABSOLUTE VALUE, AND INEQUALITY SYMBOLS

- CHAPTER 5: FACTORS, COMMON FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS

- CHAPTER 6: PROPERTIES OF NUMBERS

- CHAPTER 7: THE WORLD OF FRACTIONS

- CHAPTER 8: EXPONENTS

- CHAPTER 9: ROOTS

- CHAPTER 10: ALGEBRAIC EXPRESSIONS

- CHAPTER 11: CARTESIAN COORDINATE SYSTEM

- CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

- CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS

- CHAPTER 14: X-INTERCEPT(ZERO) AND Y INTERCEPT(B)

- CHAPTER 15: LINES

- CHAPTER 16: FUNCTIONS

- CHAPTER 17: MULTIPLYING POLYNOMIALS

- CHAPTER 18: FACTORING

- CHAPTER 19: RATIONAL EXPRESSIONS

- CHAPTER 20: SOLVING EQUATIONS

- CHAPTER 21:SOLVING INEQUALITIES

- CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS

- CHAPTER 23: QUADRATICS

- CHAPTER 24: CIRCLES

- CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES

- CHAPTER 26: VOLUMES