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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

In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
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the polar co-ordinate of the point (-1, -1)
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prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
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tanh`(x-iy) =A+iB, find A and B
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B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
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the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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1+cos²A/cos²A=2cosec²A-1
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test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
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answer
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exponential series
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what is subgroup
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Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
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e power cos hyperbolic (x+iy)
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10y
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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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