|Real Number System |||Absolute Value |||Functions and Graphs |||Trigonometry |||Basic Graphing Skills |||Studying a Function |||Algebraic Skills |||Answer Section|
|Section 1 | Section 2 | Section 3 |||Section 4 | Section 5 |||Section 6 | Section 7 | Section 8 | Section 9 | Section 10 | Section 11 | Section 12 |||Section 13 | Section 14 | Section 15 | Section 16 | Section 17 | Section 18 |||Section 19 |||Section 20 | Section 21 | Section 22 |||Section 23 |||Answers|
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.
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:
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.
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)
sign of (x-5)
sign of (x+2)(x-5) or the product of the two factors:
(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.
(This problem can be solved similar to example 3 after 1 is subtracted from each side of the inequality statement.)
To determine where this is true complete the following sign studies:
sign of (x-3)
sign of (x-2)
sign of or the quotient of the two factors:
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.)
This page was modified on 03/21/16
© Rahn, 2000