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Section exercises

Verbal

When solving an inequality, explain what happened from Step 1 to Step 2:

Step 1 2 x > 6 Step 2 x < 3

When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.

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When solving an inequality, we arrive at:

x + 2 < x + 3 2 < 3

Explain what our solution set is.

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When writing our solution in interval notation, how do we represent all the real numbers?

( , )

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When solving an inequality, we arrive at:

x + 2 > x + 3 2 > 3

Explain what our solution set is.

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Describe how to graph y = | x 3 |

We start by finding the x -intercept, or where the function = 0. Once we have that point, which is ( 3 , 0 ) , we graph to the right the straight line graph y = x −3 , and then when we draw it to the left we plot positive y values, taking the absolute value of them.

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Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.

3 x + 2 7 x 1

( , 3 4 ]

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4 ( x + 3 ) 2 x 1

[ 13 2 , )

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1 2 x 5 4 + 2 5 x

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−5 ( x 1 ) + 3 > 3 x 4 4 x

( , 3 )

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−3 ( 2 x + 1 ) > −2 ( x + 4 )

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x + 3 8 x + 5 5 3 10

( , 37 3 ]

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x 1 3 + x + 2 5 3 5

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For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.

| x + 9 | −6

All real numbers ( , )

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| 3 x 1 | > 11

( , 10 3 ) ( 4 , )

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| x 2 | + 4 10

( , −4 ] [ 8 , + )

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| x 7 | < −4

No solution

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| x 3 4 | < 2

( −5 , 11 )

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For the following exercises, describe all the x -values within or including a distance of the given values.

Distance of 5 units from the number 7

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Distance of 3 units from the number 9

[ 6 , 12 ]

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Distance of10 units from the number 4

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Distance of 11 units from the number 1

[ −10 , 12 ]

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For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

3 x + 1 > 2 x 5 > x 7

x > 6  and  x > 2 Take the intersection of two sets . x > 2 , ( 2 , + )

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2 x 5 < −11     or     5 x + 1 6

x < 3 or x 1 Take the union of the two sets . ( , 3 ) [ 1 , )

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Graphical

For the following exercises, graph the function. Observe the points of intersection and shade the x -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

| x 1 | > 2

( , −1 ) ( 3 , )


A coordinate plane where the x and y axes both range from -10 to 10.  The function |x  1| is graphed and labeled along with the line y = 2.  Along the x-axis there is an open circle at the point -1 with an arrow extending leftward from it.  Also along the x-axis is an open circle at the point 3 with an arrow extending rightward from it.

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| x + 7 | 4

[ −11 , −3 ]


A coordinate plane with the x-axis ranging from -14 to 10 and the y-axis ranging from -1 to 10.  The function y = |x + 7| and the line y = 4 are graphed.  On the x-axis theres a dot on the points -11 and -3 with a line connecting them.

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| x 2 | < 0

It is never less than zero. No solution.


A coordinate plane with the x and y axes ranging from -10 to 10.  The function y = |x -2| and the line y = 0 are graphed.

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For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y -values of the lines.

x 2 > 2 x + 1

Where the blue line is above the orange line; point of intersection is x = 3.

( , −3 )


A coordinate plane with the x and y axes ranging from -10 to 10.  The lines y = x - 2 and y = 2x + 1 are graphed on the same axes.

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1 2 x + 1 > 1 2 x 5

Where the blue line is above the orange line; always. All real numbers.

( , )


A coordinate plane with the x and y axes ranging from -10 to 10.  The lines y = x/2 +1 and y = x/2  5 are both graphed on the same axes.

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Numeric

For the following exercises, write the set in interval notation.

{ x | −1 < x < 3 }

( −1 , 3 )

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{ x | x < 4 }

( , 4 )

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{ x | x  is all real numbers }

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For the following exercises, write the interval in set-builder notation.

( , 6 )

{ x | x < 6 }

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[ −3 , 5 )

{ x | −3 x < 5 }

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[ −4 , 1 ] [ 9 , )

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For the following exercises, write the set of numbers represented on the number line in interval notation.

Questions & Answers

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
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
how do they get the third part x = (32)5/4
kinnecy Reply
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
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
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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