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Using interval notation to express all real numbers less than or equal to a Or greater than or equal to b

Write the interval expressing all real numbers less than or equal to −1 or greater than or equal to 1.

We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at and ends at −1 , which is written as ( , −1 ] .

The second interval must show all real numbers greater than or equal to 1 , which is written as [ 1 , ) . However, we want to combine these two sets. We accomplish this by inserting the union symbol, , between the two intervals.

( , −1 ] [ 1 , )
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Express all real numbers less than −2 or greater than or equal to 3 in interval notation.

( , −2 ) [ 3 , )

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Using the properties of inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

Properties of inequalities

A d d i t i o n   P r o p e r t y If  a < b ,  then  a + c < b + c . M u l t i p l i c a t i o n   P r o p e r t y If  a < b  and  c > 0 ,  then  a c < b c . If  a < b  and  c < 0 ,  then  a c > b c .

These properties also apply to a b , a > b , and a b .

Demonstrating the addition property

Illustrate the addition property for inequalities by solving each of the following:

  • (a) x 15 < 4
  • (b) 6 x 1
  • (c) x + 7 > 9

The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.


  1. x 15 < 4 x 15 + 15 < 4 + 15   Add 15 to both sides . x < 19

  2. 6 x 1 6 + 1 x 1 + 1 Add 1 to both sides . 7 x

  3. x + 7 > 9 x + 7 7 > 9 7 Subtract 7 from both sides . x > 2
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Solve: 3 x −2 < 1.

x < 1

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Demonstrating the multiplication property

Illustrate the multiplication property for inequalities by solving each of the following:

  1. 3 x < 6
  2. −2 x 1 5
  3. 5 x > 10

  1. 3 x < 6 1 3 ( 3 x ) < ( 6 ) 1 3 x < 2

  2. 2 x 1 5 2 x 6 ( 1 2 ) ( 2 x ) ( 6 ) ( 1 2 ) Multiply by  1 2 . x 3 Reverse the inequality .

  3. 5 x > 10 x > 5 ( 1 ) ( x ) > ( 5 ) ( 1 ) Multiply by  1. x < 5 Reverse the inequality .
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Solve: 4 x + 7 2 x 3.

x −5

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Solving inequalities in one variable algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

Solving an inequality algebraically

Solve the inequality: 13 7 x 10 x 4.

Solving this inequality is similar to solving an equation up until the last step.

13 7 x 10 x 4 13 17 x −4 Move variable terms to one side of the inequality . −17 x −17 Isolate the variable term . x 1 Dividing both sides by  −17  reverses the inequality .

The solution set is given by the interval ( , 1 ] , or all real numbers less than and including 1.

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Solve the inequality and write the answer using interval notation: x + 4 < 1 2 x + 1.

( 2 , )

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Solving an inequality with fractions

Solve the following inequality and write the answer in interval notation: 3 4 x 5 8 + 2 3 x .

We begin solving in the same way we do when solving an equation.

3 4 x 5 8 + 2 3 x 3 4 x 2 3 x 5 8 Put variable terms on one side . 9 12 x 8 12 x 5 8 Write fractions with common denominator . 17 12 x 5 8 x 5 8 ( 12 17 ) Multiplying by a negative number reverses the inequality . x 15 34

The solution set is the interval ( , 15 34 ] .

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Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
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I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
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hii
Uday
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Ali
greetings from Iran
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salut. from Algeria
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Nharnhar
Practice Key Terms 4

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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