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Chapter 6
Solving and Graphing Inequalities
Rules for graphing your answersIf the letter is on the left then we can follow
the direction of the arrowWe must mark the numbers
< or > we use an open circle< or > we use a closed circle
Our answers will look like this and we will graph the answers on a number line
1. X < 22. X > -23. Z > 14. 0 < d
Now you practice:1. T > 12. X >-13. N < 04. 4 >y
Our problems will look like this:
1. X + 5 > 3
We will use our Chapter 3 rules to solve:1. Draw tracks2. Count variables
1. If one, numbers jump tracks
2. If 2 or more where are they1. Same side, family
members2. Different sides, letters
jump3. Add or subtract the
numbers4. Is there a number on the
letters? 1. If its an integer divide
each side2. If it’s a fraction, flip and
multiply3. if the integer or fraction
is negative I must turn the arrow around to the opposite direction
5. Graph the solution
More examples:1. X + 4 < 72. N + 6 > 23. 5 > a + 54. -2 > n – 45. X – 5 > 26. P – 1 < -47. -3 < y - 2
Pg 327 #’s 42 – 54 even
Sec 6.2Solving Equations using Multi or DivWe must be able to identify if there is an
integer on the letter or a fraction on the letter.
If the integer or the fraction is negative then we MUST change the direction of the arrow in our answer
Examples1. a/4 < 42. 4x > 203. k/4 < ½4. 18 < 2k5. 6 < t/56. -21 < 3y
Examples: Negative numbers1. -1/2 y < 52. -12m > 183. -8x < 204. -1/5 p > 15. -2/3 x < -56. -24 < 6t
Word Problems1. Kayla wants to buy some posters for her
dorm room. Posters are on sale for $6 each. Write and solve an inequality to determine how many posters she can buy and spend no more than $25.
2. Crandell plans to take figure skating lessons. He can rent skates for $5 per lesson. He can buy skates for $75. For what number of lessons is it cheaper for him to buy rather than rent skates?
Pg 334 #’s 36 – 46 even
QuizSections 6.1 and 6.2
1.X – 4 < -52.-11 > y + 43.-1/2 x > -54.-3x < -275.-3/4 x < -1/4
Sec 6.3; Solving Multi-step Inequalities
1. 2y – 5 < 72. 5 – x > 43. 3(x + 2) <
74. -2(x + 1) <
25. 2x – 3 > 4x
– 1
Sec 6.3; Solving Multi-step Inequalities
1. 5n – 21 < 8n2. -3z + 15 > 2z3. X + 3 > 2x – 44. 4y – 3 < -y + 12
Word ProblemsYou plan to make and sell candles. You pay
$12 for instructions. The materials for each candle cost $0.50. You plan to sell each candle for $2. Let x be the numbers of candles you sell. How many candles must be sold to make at least $300 profit.
Sec 6.4; Solving compound inequalitiesHow do you combine two thoughts in
English class?What are they called?What words do we use?
How to Solve:1. -2 < x + 2
< 42. -1 < x + 3
< 73. -6 < -3x <
124. 0 < x – 4 <
12
These are “AND” problems. These problems must be re-written into two problems.
Word Problems1. In the summer it took a Pony Express rider
about 10 days to ride from St. Joseph, Missouri to Sacramento, California. In winter it took as many as 16 days. Write an inequality to describe the number of days that the trip might have taken.
Word ProblemsFrequency is used to describe the pitch of a
sound. Frequencies are measured in hertz. Write an inequality for the followingSound of a human voice: 85 hertz to 1100
hertzSound of a bats signal: 10,000 hertz to
120,000 hertzSound heard by a dog: 15 hertz to 50,000
hertzSound heard by a dolphin: 150 hertz to
150,000 hertz
Pg 346 #’s 30-46 even
Sec 6.5; Solving Compound InequalitiesThese are called “OR” problemsThese problems are already written and
ready to be solved
Examples1. X-4<3 or 2x> 182. 3x + 1 < 4 or 2x – 5 > 73. X + 5 < -6 or 3x > 124. 6x – 5 < 7 or 8x + 1 > 25
Word ProblemA baseball is hit straight up in the air. Its
initial velocity is 64 ft per second. Its formula is v = -32t + 64. Find the values of “t” for which the velocity of the baseball is greater than 32 or less than -32 feet per second.
Sec 6.6; Absolute Value EquationsThese problems will create some re-writing
1st – make sure the absolute value bars are on a side by themselves
2nd – drop the bars and write two problems3rd – in the second problem you will need the
opposite of the symbol (equals or inequality) and the opposite of the number.
Now solve the equationsIf its an inequality then you will also graph
the solutions.
For examples we will use p. 356
Pg. 358 #”s 16-26 evenpg 359 #’s 32-40 even
Sec 6.7; Solving absolute value inequalitiesWe will use the same rules as the previous
section (6.6)Make sure to remember to make the
changes to the second problem that you re-write.
We will use pg 364 and 365 for examples
Sec 6.8; Graphing Linear Inequalities in Two VariablesWe use our previous knowledge from
Chapter’s 4 and 5.
To graph we will use y= mx + bSlope intercept form: y = mx + b
“b” is the y-intercept. I must always use this number first when graphing. It is always located on the y-axis, either above or below the origin.
“m” is the slope. Its always a fraction and remember to use rise over run
To graph given when given an equation; turn it into y=mx+bIf the symbol is < or
> we will draw a dashed line
If the symbol is < or > we will draw a normal line
To shade will have to use a test point and the slope intercept form
If the test result is false shade away from the TP, if the test is true shade to the TP.
Flow ChartLet x jump the
tracksIf there is a
number on y, then we will set up three fractions and divide or reduce
Collect “b” and find it on the y-axis
Collect “m” and use rise over run to get to the next point on my line.
-2x + y < 3
MORE EXAMPLES:1. X < -22. Y < 13. X + Y > 34. 2X – Y > -25. 3X – Y < 4
P 371 #’S 26-30 EVEN
P 371 #’S 36-50 EVEN
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