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Solve the following:Solve the following:(8 + v)2 – 10 = 22(8 + v)2 – 10 = 22
Sec 6.5 Solving Inequalities Using Addition and Subtraction
1. graph inequalities on a number line.
2. solve inequalities using addition and subtraction.
6-5 Solving Inequalities 6-5 Solving Inequalities by Adding & Subtractingby Adding & Subtracting
ObjectiveThe student will be able to:
To solve for a variable the To solve for a variable the variable must be positive with variable must be positive with
no number beside it!! no number beside it!! x = 10 Solved
- x = -5 Not solved
2x = 5 Not solved
-3x =6 Not solved
INEQUALITIESAnother type of sentence used in algebra is called an inequality.
An inequality is used when we don't know exactly what an expression is equal to. Instead of an equals sign, we use one of
these symbols:This will be easier to do if you remember that the open part
of the symbol always faces the larger quantity.
SYMBOLSYMBOL MEANINGMEANING
< < less thanless than
>> greater thangreater than
≤ ≤ less than or equal toless than or equal to
≥ ≥ greater than or equal togreater than or equal to
Graph the solution set ofx < 3
o
When you have < or >, use an open dotopen dot! 0 5-5
Graph the solution set ofy ≥ -5/4
•
When you have ≤ or ≥, use a closed dotclosed dot! Convert -5/4 to a decimal = -1.25
0 5-5
Graph the solution set ofp ≠ 2
o
When you have ≠, use an open dotopen dot and shade both ways!
0 5-5
What is the graph of 7 ≤ m?
o7 86o7 86●7 86
7 86●
1.
2.
3.
4.
Answer NowAnswer Now
What type of dot open or closed?
1. <
2. >
3. ≤
4. ≥
5. ≠
Open
Open
Closed
Closed
Open
Addition & Subtraction property of inequality
• You can add or subtract the same number or term on each side of the inequality, the relationship between the two sides does not change.
y + 21 ≥≥ 7
y + 21 - 21 ≥≥ 7 - 21
x - 14 << 16
x – 14 + 14 << 16 +14
1) Solve x + (-14) < 16 x - 14 < 16
+ 14 + 14
x < 30
30 + (-14) = 16
16 = 16
Solve this problem like an equation
1. Draw “the river”2. Eliminate double
signs3. Add 14 to both sides4. Simplify5. Check your answer6. Graph the solution
o30 3129
2) Solve y + 21 ≥ 7 - 21 -21
y ≥ -14
30 + (-14) = 16
16 = 16
1. Draw the “river”2. Subtract 21 from
both sides3. Simplify4. Check your answer5. Graph the solution
-14 -13-15●
3) Solve 8y + 3 > 9y - 14
o17 1816
- 8y - 8y
3 > y - 14
+ 14 + 14
17 > y
y < 178(17) + 3 = 9(17) - 14
1. Draw “the river”
2. Subtract 8y from both sides
3. Simplify
4. Add 14 to both sides
5. Simplify
6. Rewrite inequality with the variable first
7. Check your answer
8. Graph the solution
4) Solve 3r - 17 ≥ 2r + 14
●31 3230
- 2r - 2r
r – 17 ≥ 14
+ 17 + 17
r ≥ 31 3(31) - 17 = 2(31) + 14
1. Draw “the river”
2. Subtract 2r from both sides
3. Simplify
4. Add 17 to both sides
5. Simplify
6. Check your answer
7. Graph the solution
Solve and graph -2x + 6 ≥ 3x - 4
x ≤ 2
0 5-5
Joanna’s tests were 87, 93, 88 and 94. What must her 5th grade be to
get a total of at least 459?
X + 87 + 93 + 88 + 94 ≥ 459
X + 362 ≥ 459 -362 -362
X ≥ 97
6-6 Solving Inequalities by Multiplying or Dividing
ObjectiveThe student will be able to:
solve inequalities using multiplication and division.
POSITIVE NUMBERS ONLYPOSITIVE NUMBERS ONLYMultiplication & Division properties of inequalities
You can multiply or divide the same positive number or positive term on each side of the inequality, the relationship between the two sides does not change.
x
3
3 10 33
x
< -108p ≥ -96
8 96
8 8
p
≥The inequality The inequality stays the samestays the same
NEGATIVE NUMBERS ONLYNEGATIVE NUMBERS ONLYMultiplication & Division properties of inequalities
When multiplying or dividing the same negative number or negative term on each side of the inequality, you must reverse the inequality sign
3x
3 10 33
x
> -10-8p ≥ -96
8 96
8 8
p
≤≤<< -3
The inequality gets flipped The inequality gets flipped every time you * or / by a every time you * or / by a
negative numbernegative number
To solve for a variable the To solve for a variable the variable must be positive with variable must be positive with
no number beside it!! no number beside it!! x = 10 Solved
- x = -5 Not solved
2x = 5 Not solved
-3x =6 Not solved
1) Solve > 13k
4
o-52 -51-53
k < -52
1. Draw “the river”
2. Clear the fraction - Multiply both sides by -4
3. NEW STEP!! When multiplying BOTH sides by a NEGATIVE number, SWITCH the inequality!
4. Simplify
5. Check your answer
6. Graph the solution
4 13 -44
k
152
34
2) When solving < -10will the inequality switch?
1. Yes!
2. No!
3. I still don’t know!
x
3
Answer NowAnswer Now
2) Solve < -10 x
3
o-30 -29-31
x < -30
1. Draw “the river”
2. Clear the fraction - Multiply both sides by 3. Do you switch the inequality?
No - Both sides are being multiplied by a positive number
3. Simplify
4. Check your answer
5. Graph the solution
3 10 33
x
030
13
3) When solvingwill the inequality switch?
1. Yes!
2. No!
3. I still don’t know!
a
4 6
Answer NowAnswer Now
3) Solve a
4 6
o-24 -23-25
a > -24
1. Draw “the river”
2. Clear the fraction - Multiply both sides by -4. Do you switch the inequality?
Yes - Both sides are being multiplied by a negative number
3. Simplify
4. Check your answer
5. Graph the solution
-4 6 -44
a
24( )6
4
4) Solve -8p ≥ -961. Draw “the river”
2. Divide both sides by -8. Do you switch the inequality?
Yes - Both sides are being divided by a negative number
3. Simplify
4. Check your answer
5. Graph the solution
8 96
8 8
p
p ≤ 12
-8(12) = -96
●12 1311
o-4 -3-5o-4 -3-5●-4 -3-5
-4 -3-5●
1.
2.
3.
4.
Answer NowAnswer Now
1) Which graph represents the correct answer to > 1k
4
Multiply both sides by -4
Switch inequality signK < -4
o-15 -14-16o-15 -14-16●-15 -14-16
-15 -14-15●
1.
2.
3.
4.
Answer NowAnswer Now
5) Solve 7v < -105
Divide both sides by 7
V < -15
Homework
Practice Sheet 6-5 all (9-14 show work)
Practice Sheet 6-6 all (show work for all)
Homework
Pg. 284 #8-26 Even
8-18 show work, graphing, and solution (3pts)
20-22, 26 write inequality solution, and the graph(3 pts)
24 just graph (1pt)
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