Home | | |||||||
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 |
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:
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)
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.
4.
(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