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Fill in < , > , or = for each of the following pairs of numbers:

| −5 | ___ | −5 | 8 ___ | −8 | −9 ___ | −9 | ( −16 ) ___ | −16 |

Solution


| −5 | ___ | −5 | Simplify. 5 ___ 5 Order. 5 > −5 | −5 | > | −5 |


8 ___ | −8 | Simplify. 8 ___ 8 Order. 8 > −8 8 > | −8 |


9 ___ | −9 | Simplify. −9 ___ 9 Order. −9 = −9 −9 = | −9 |


( −16 ) ___ | −16 | Simplify. 16 ___ 16 Order. 16 > −16 ( −16 ) > | −16 |

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Fill in<,>, or = for each of the following pairs of numbers: | −9 | ___ | −9 | 2 ___ | −2 | −8 ___ | −8 |
( −9 ) ___ | −9 | .

> > < >

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Fill in<,>, or = for each of the following pairs of numbers: 7 ___ | −7 | ( −10 ) ___ | −10 |
| −4 | ___ | −4 | −1 ___ | −1 | .

> > > <

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We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value    of the resulting number.

Grouping symbols

Parentheses ( ) Braces { } Brackets [ ] Absolute value | |

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

Simplify: 24 | 19 3 ( 6 2 ) | .

Solution

24 | 19 3 ( 6 2 ) | Work inside parentheses first: subtract 2 from 6 . 24 | 19 3 ( 4 ) | Multiply 3 ( 4 ) . 24 | 19 12 | Subtract inside the absolute value bars. 24 | 7 | Take the absolute value. 24 7 Subtract. 17

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Simplify: 19 | 11 4 ( 3 1 ) | .

16

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Simplify: 9 | 8 4 ( 7 5 ) | .

9

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Evaluate: | x | when x = −35 | y | when y = −20 | u | when u = 12 | p | when p = −14 .

Solution

| x | when x = −35

| x |
. .
Take the absolute value. 35


| y | when y = −20
| y |
. .
Simplify. | 20 |
Take the absolute value. 20


| u | when u = 12
| u |
. .
Take the absolute value. 12


| p | when p = −14
| p |
. .
Take the absolute value. 14

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Evaluate: | x | when x = −17 | y | when y = −39 | m | when m = 22 | p | when p = −11 .

17 39 −22 −11

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Evaluate: | y | when y = −23 | y | when y = −21 | n | when n = 37 | q | when q = −49 .

23 21 −37 −49

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Add integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

Doing the Manipulative Mathematics activity “Addition of Signed Numbers” will help you develop a better understanding of adding integers.”

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.

We will use the counters to show how to add the four addition facts using the numbers 5 , −5 and 3 , −3 .

5 + 3 −5 + ( −3 ) −5 + 3 5 + ( −3 )

To add 5 + 3 , we realize that 5 + 3 means the sum of 5 and 3.

We start with 5 positives. .
And then we add 3 positives. .
We now have 8 positives. The sum of 5 and 3 is 8. .

Now we will add −5 + ( −3 ) . Watch for similarities to the last example 5 + 3 = 8 .

To add −5 + ( −3 ) , we realize this means the sum of −5 and 3 .

We start with 5 negatives. .
And then we add 3 negatives. .
We now have 8 negatives. The sum of −5 and −3 is −8. .

In what ways were these first two examples similar?

  • The first example adds 5 positives and 3 positives—both positives.
  • The second example adds 5 negatives and 3 negatives—both negatives.

Questions & Answers

4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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