Chapter 4 Study Guide /Notes
4.1 Inequalities and Their Graphs pp 200-202
Terms
A solution of an inequality - any number that makes the inequality true.
○ use when your inequality has the symbol or >. This means that the inequality cannot equal the endpoint. x < 3
use when your inequality has the symbol ≤ or ≥. This means that the inequality can equal the endpoint. x ≤ 3
If x is on the left the greater than or less than sign will point the way the line should go!
4.2 Solving Inequalities Using Addition and Subtraction pp 206-208
Terms
Equivalent inequalities – inequalities with the same solutions.
Addition Property of Inequalities - If a > b, then a + c > b + c. If 6 > 2, then 6 + 5 > 2 + 5 (11 > 7).
Subtraction Property of Inequalities - If a > b, then a - c > b - c. If 7 > 5, then 7 – 3 > 5 – 3 (4 > 2).
You can solve inequalities using the same addition and subtraction methods you used for solving equations.
x – 6 < 8 add 6 to both sidesy + 5 ≥ 12 subtract 5 from both sides
+6 +6 - 5 - 5
x < 14 y ≥ 7
4.3 Solving Inequalities Using Multiplication and Division pp 212-215
Terms
Multiplication Property of Inequality (for positive numbers)
If a > b, thena•cb•c. If 10 > 5, then 10 • 4 > 5 • 4 (40> 20)
Multiplication Property of Inequality (for negative numbers)
If a > b, thena•cb•c. If 10 > 5, then 10 •(- 4) < 5 •(-4) (-40<- 20)
Division Property of Inequality (for positive numbers)
If a > b, then> . If 15 > 10, then > (3 > 2)
Division Property of Inequality (for negative numbers)
If a > b, then< . If 15 > 10, then (- 3 - 2)
4.4 Solving Multi-Step Inequalities pp 219-221
Remembering the rules for solving inequalities are much like the rules for solving equations with the exception of multiplying or dividing by a negative!
5x – 3(x -2) < 6x – 10distribute
5x – 3x + 6 < 6x – 10combine like terms on each side of the inequality sign
2x + 6 < 6x – 10get the variables on one side
-6x -6x
-4x + 6 < -10the constants on the other
-6 -6
-4 x-16 divide by -4 --- don’t forget to flip the sign!!!
-4 -4
x > 4Check a number greater than 4 to see if you solved correctly.
5(8) – 3((8) -2) < 6(8) – 10
40 – 3 (6) < 48 – 10
40 – 18 < 38
22 < 38 Correct!
4.5 Compound Inequalities pp 227-229
Terms
Compound Inequality – inequalities than are joined by the words and or or.
x < 5 and x ≥ -2
“and “ means both inequalities must be true, so the overlapping parts of the lines are the solution.
The inequality can be written -2 ≤ x < 5.
x < -2 or x ≥ 1
“or “ means either inequalities can be true, so all the parts of both lines are the solution.
4.6 Absolute Value Equations and Inequalities pp 235-237
Absolute Value Equations
To solve an absolute value equation, first get just the absolute value portion of the equation on one side.
6 + |2x – 3| = 13subtract 6
-6 -6
|2x – 3| = 7
2x – 3 could equal either 7 or – 7 so set up two equations from the original one.
2x – 3 = 72x – 3 =- 7
+3 +3add 3 to both sides +3 +3
2x = 10 2x = -4
2 2 divide by 2 2 2
x = 5 or x = -2
Check your work!
6 + |2(5) – 3| = 13 6 + |2(-2) – 3| = 13
6 + |10 - 3| = 13 6 + |(-4) – 3| = 13
6 + |7| = 13 6 + |-7| = 13
6 + 7 = 13 6 + 7 = 13
13 = 13 13 = 13
Absolute Value Inequalities
To solve an absolute value inequality, first get just the absolute value portion of the inequality on one side.
3 + |5x – 10| < 13subtract 3
-3 -3
|5x – 10| < 10
5x -10 must be less than 10 and greater than -10. (Between -10 and 10)
5x – 10 < 105x – 10 > - 10
+10 +10add 10 to both sides +10 +10
5x20 5x0
5 5 divide by 5 5 5
x< 4 and x > 0
Check your work! A number that is less than 4 and greater than 0 would be 1.
( 2 or 3 would also work.)
3 + |5(1) – 10| < 13
3 + |5 - 10| < 13
3 + |-5| < 13
3 + 5 < 13
8 < 13 True!
Here is another:
2 + |3x + 15| > 8subtract 2
-2 -2
|3x + 15| > 6
3x + 15 must be bigger than 6 or smaller than -6.
3x + 15 > 63x + 15 < - 6
-15 -15 subtract 15 -15 -15
3x > -9 3x < -21
3 3 divide by 3 3 3
x > -3 or x < -7
Check with 0!Check with -10!
2 + |3(0) + 15| > 82 + |3(-10) + 15| > 8
2 + |0 + 15| > 8 2 + | -30 + 15| > 8
2 + |15| > 8 2 + |-15|> 8
2 + 15 > 8 2 + 15 > 8
17 > 8True! 17 > 8