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(Section 5)

1.  means b - a is positive.

2.   means  or .

3. If , and  , then .

4. If  , then  and .

5. If , then  when c is positive.

6. If  then when c is negative.

7. If  and  , then .

8. If a and b are both positive or both negative and  , then .

9. If , then x is a positive number (or non-negative).

10. If , then x is a non-positive number (zero or negative).

Knowing how to solve inequalities is an important skill. The first two sample problems demonstrate how linear inequalities are solved using the rules listed above.
 

Sample Problems:
 

1. 3x + 5 > 2x - 5

        x > -10

Note that the original inequality 3x+5>2x-5 is equivalent to the inequality x>-10. But this is also equivalent to x+10>0. Visually this looks like this:

number line

This number line is showing that the expression x+10 is positive when x >10 , negative when x<-10, and zero when x = -10.

Another way to think about this is to think about the line y = x+10. Where does this line pass through the x-axis? 

Where is this line below the x-axis? Where is this line above the x-axis? The answer to these three questions will give you the same sign study. 

When the line y=x+10 is below the x-axis, the expression x+10 is negative. When the line y=x+10 is above the x-axis, the expression x+10 is positive. When the line y=x+10 passes through the x-axis, the expression x+10 is zero.

2. -5x + 2 > -8

        -5x > -10

        x < 2

Again note that the original inequality -5x+2>-8 is equivalent to x<2. Remember that this is also equivalent to x-2<0. Visually this looks like this:

The number line is showing that the expression x-2 is negative when x<2, positive when x>2 and zero when x = 2. You can create this sign study by thinking about the line y=x-2 as we did in the previous example.

3. (x+2)(x-5)>0

The third sample problem is a little more difficult to analyze. This inequality means that the product of two factors must be positive. This can happen two ways: both factors are positive or both factors are negative.

sign of (x+2)

number line
 

sign of (x-5)

number line

sign of (x+2)(x-5) or the product of the two factors:

 number line

(You can also build the third number line of signs by first selecting the two places where x+2 and x-5 each equal zero. Place these on the number line and then test sample values in each region to decide the sign of (x+2)(x-5).
 

So the solution is x>5 OR x<-2 because this is where the product of (x+2) and (x-5) is positive. (Be careful not to say the solution is x>5 and x<-2.) This answer cannot be combined into a triple inequality (The triple inequality 5<x<-2 is really an empty set.) It must be written as two inequalities joined together with the conjunction OR.

4. number line

(This problem can be solved similar to example 3 after 1 is subtracted from each side of the inequality statement.)

    solution
 

    solution
 

    solution

To determine where this is true complete the following sign studies:

sign of (x-3)

 number line

sign of (x-2)

 number line

sign of  or the quotient of the two factors:

number line
 
Again you can create this third line by simply placing a zero above the two locations x-3 and x-2 each equal zero. Then test sample points in each region to decide the sign of .  Note that when x<2 or x>3 the fraction is negative or less than zero. You must exclude x=2 because x-2 is an expression in the denominator and a denominator of a fraction cannot equal zero.
 

So the solution is  (This is the same as saying x>2 AND  .)
 

Check Your Understanding: (Remember to write down your answers.)
 

  1. Which of the rules listed above are new to you? Explain what each of these says.
     
  2. Give a numerical example of each rule.
  3. Find all values of x where 2 - 5x > 7 and 3x + 1 > -11.
     
  4. Find all values of x where 2x + 3 < 4 or 3x + 3 > 9.
     
  5. Find all values of x where (x+3)(2x-1)<0
     
  6. Find all values of x where 

 

  1. Find all values of x where .

 

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Section 4

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This page was modified on 03/21/16

Rahn, 2000