Even More Intermediate
Algebra
Problems and Solutions
with your fx-991ES or fx-
115ES Calculator
Dr Allen Brown
Cambridge
Paperbacks
Cambridge Paperbacks
www.CambridgePaperbacks.com
First published by Cambridge Paperbacks 2018
© Allen Brown 2018
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system without
permission in writing from the author.
Disclaimer
Although the author and publisher have made every effort to ensure that the
information in this book was correct during preparation and printing, the
author and publisher hereby disclaim any liability to any party for any errors
or omissions.
Read this First
Continuing the theme from the previous two ebooks
on Intermediate Algebra Problems and Solutions, this
ebook has more examples showing you how to make
effective use of your fx-991ES or fx-115ES calculator.
The emphasis has been made on re-arranging
expressions in order to perform calculations to
determine the value of x. Once x has been
determined, the SOLVE feature in the fx calculator has
been used to confirm the answer.
This is very powerful technique for you to acquire in
order to ensure the correctness of your result. You are
encouraged to use it when solving algebraic problems
on your course of study whether it’s at GCSE level or
higher.
As you progress working through this ebook your
understanding of maths will increase and as stated
previously, it’s therefore important you perform all
the keystrokes yourself to maximise the learning
experience.
Dr Allen Brown
Cambridgeshire
Contents
1 Equations with Reciprocals ................................................................. 3
Example 1-1: ....................................................................................... 3
Example 1-2: ....................................................................................... 6
Example 1-3: ....................................................................................... 6
Example 1-4: ....................................................................................... 7
Example 1-5: ....................................................................................... 8
Example 1-6: ....................................................................................... 9
Example 1-7: ..................................................................................... 10
2 Quadratic Expressions ....................................................................... 12
Example 2-1: ..................................................................................... 12
Example 2-2: ..................................................................................... 13
Example 2-3: ..................................................................................... 14
Example 2-4: ..................................................................................... 15
Example 2-5: ..................................................................................... 16
Example 2-6: ..................................................................................... 17
3 Cubic Expressions .............................................................................. 19
Example 3-1: ..................................................................................... 19
Example 3-2: ..................................................................................... 21
Example 3-3: ..................................................................................... 23
4 Trigonometric Expressions ................................................................ 25
Example 4-1: ..................................................................................... 25
Example 4-2: ..................................................................................... 27
1
Example 4-3: ..................................................................................... 29
Example 4-4: ..................................................................................... 30
Example 4-5: ..................................................................................... 31
Example 4-6: ..................................................................................... 33
5 Logarithmic and Exponential Expressions ........................................ 35
Example 5-1: ..................................................................................... 35
Example 5-2: ..................................................................................... 37
Example 5-3: ..................................................................................... 39
Example 5-4: ..................................................................................... 40
6 Indices ................................................................................................ 43
Example 6-1: ..................................................................................... 43
Example 6-2: ..................................................................................... 45
Example 6-3: ..................................................................................... 46
Example 6-4: ..................................................................................... 47
Example 6-5: ..................................................................................... 49
Example 6-6: ..................................................................................... 50
7 Changing the Subject of an Equation................................................ 52
Example 7-1: ..................................................................................... 52
Example 7-2: ..................................................................................... 55
Example 7-3: ..................................................................................... 56
Example 7-4: ..................................................................................... 58
2
1 Equations with Reciprocals
Very often you will come across equations which have
fractional components and you are asked to get all the
x values onto the left hand side of the equation.
Following the golden rule, whatever you do on one
side of the equation you need to do the equivalent on
the other side. Your fx-991ES calculator is really useful
for indicating whether you have the right answer
using the SOLVE feature. In the following examples,
the keystrokes will be given for each equation.
Example 1-1:
Solve the equation,
− 5
+ = 10
4 3
With the 4 and 3, the common value is 12, need to
multiple both sides by 12,
− 5
12 × + 12 × ( ) = 12 × 10
4 3
This becomes,
3 + 4( − 5) = 120
3
Expand the bracket,
3 + 4 − 20 = 120
Add 20 to both sides of the equation,
3 + 4 − 20 + 20 = 120 + 20
Leaving
7 = 140
Divide both sides by 7,
7 140
=
7 7
Giving the result = 20. This result can be confirmed
using the SOLVE feature on your fx calculator. Here is
the original equation,
− 5
+ = 10
4 3
Enter the following keystrokes, (X)
CQ(X)a4$+aJ(X)p5R3
$Q(=)10
4
Notice we use the Q= not the = in the bottom
right corner on your calculator. Now enter,
q(SOLVE) and you see in the display,
You are now expected to enter a guess value for X, this
is how the algorithm in the calculator works – it needs
a seed value (a good guess). Enter 1=,
You will see the correct value for X = 20 which is what
we previously derived. The L-R, indicates whether
there was any difference between the left hand side
and the right hand side of the equation. Sometimes
(SOLVE) only gets an approximation for X and this
what L-R shows.
In the remaining examples you can solve the
equations yourself and we shall use SOLVE to show
whether your answer is correct.
5
Example 1-2:
Solve for x,
− 5 + 5
+ = 5
10 5
The keystrokes for solving this equation are.
CaQ(X)p5R10$+aJ(X)
+5R5$Q(=)5
q(SOLVE) 1=
x = 15.
Example 1-3:
Solve for x,
− 2 + 10
+ = 5
2 9
6
The keystrokes are,
CaQ(X)p2R2$+aJ(X)+
10R9$Q(=)5
q(SOLVE) 1=
x = 8.
Example 1-4:
Solve for x,
+ 19
= 3 +
5 4
The keystrokes are,
CaQ(X)+19R5$Q(=)3+
J(X)a4
7
q(SOLVE) 1=
x = 16.
Example 1-5:
Solve for x,
− 4 − 10
=
7 5
The keystrokes are,
CaQ(X)p4R7$Q(=)aJ(X)
p10R5
q(SOLVE) 1=
8
x = 25.
Example 1-6:
Solve for x,
+ 5 + 1 + 3
− =
6 9 4
The keystrokes are,
CaQ(X)+5R6$paJ(X)+
1R9$Q(=)aJ(X)+3R4
q(SOLVE) 1=
1
= −0.1426. . = −
7
9
Example 1-7:
Solve for x,
+
=
−
1 When A =9.716, B = 2.328, C = 7.95 and D = 1.693
2 When A = 58.97, B =39.93, C = 4.33 and D = -123.84
1 The keystrokes are,
CaQ(A)+Q(B)RQ(C)pQ(D)$
Q(=)Q(X)
q(SOLVE)
A? 9.716=
B? 2.328=
C? 7.95=
D? 1.693=
=
10
= 1.92488
2 Enter the keystrokes
q(SOLVE)
A? 58.97=
B? 39.93=
C? 4.33=
D? z123.84=
=
= 0.7716
11
2 Quadratic Expressions
When solving quadratic equations, you are already
familiar with the well expression,
2
− ± √ − 4
2
2
You will have real solutions provided ≥ 4 . To
perform this calculation, use the multi-stage features
2
of your fx calculator, let = √ − 4 first. The
keystrokes are,
CQ(F)Q(=)sQ(B)dp4Q(A)
Q(C)$Q(:)
azJ(B)+J(F)R2J(A)$Q(:)
azJ(B)pJ(F)R2J(A)
After entering all these keystrokes, this is what you
will see in the display.
Example 2-1:
1 Solve the following quadratic expression,
12
2
5 + 14 − 55
Enter the keystrokes,
r
B? 14=
A? 5=
C? z55=
= =
The expression becomes,
2
5 + 14 − 55 = (5 − 11)( + 5)
Example 2-2:
2 Solve the following quadratic expression,
2
3 − 44 + 121
Enter the keystrokes,
r
B? z44=
13
A? 3=
C? 121=
= =
The expression becomes,
2
3 − 44 + 121 = ( − 11)(3 − 11)
Example 2-3:
3 Solve the following quadratic expression,
2
6 − 25 + 21
Enter the keystrokes,
r
B? z25=
A? 6=
C? 21=
= =
14
The expression becomes,
2
6 − 25 + 21 = ( − 3)(6 − 7)
Example 2-4:
4 Solve the following quadratic expression,
2
50 − 15 − 27
Enter the keystrokes,
r
B? z15=
A? 50=
C? z27=
= =
The expression becomes,
2
50 − 15 − 27 = (10 − 9)(5 + 3)
You can of course use the EQN option from the MODE
to solve quadratic equations; these examples show
the versatility of the fx-991ES calculator.
15
Example 2-5:
5 Solve the following quadratic expression,
5 4 3
− =
− 2 + 6
After you have re-arranged this expression to look like
a quadratic you should be able to find the solutions.
We shall use the SOLVE feature on your fx-991ES
calculator. The keystrokes are,
C5aQ(X)p2$p4aJ(X)$
Q(=)3aJ(X)+6
There are two solutions, we shall use two seed values,
-3 and +3,
q(SOLVE)
Solve for X z3=
16
q(SOLVE)
Solve for X 3=
The expression can be factorised to give,
( + 2)( − 12)
This should confirm the result you got from re-
arranging the original expression.
Example 2-6:
6 Solve the following quadratic expression,
3 − 2 5
= − 2
2 − 3 + 4
More re-arranging to get it into a recognisable
quadratic form. It can be solved directly similar to the
previous example. The keystrokes for your fx
calculator are,
Ca3Q(X)p2R2J(X)p3$
Q(=)5J(X)aJ(X)+4$p2
17
q(SOLVE)
Solve for X 3=
q(SOLVE)
Solve for X z3=
You will recognise that 10.666.. = 10⅔ , the factorised
quadratic is therefore,
(3 − 32)( − 1)
18
3 Cubic Expressions
3
You will encounter equations which have x and here
are some examples of how they are solved, that is
bring all the x values onto the left hand side.
Example 3-1:
Solve for x in the following equation,
3 3
3
+ 7 = + 2
4 5
Subtract ¾ from both sides,
3 3 3 3
3
+ 7 − = + 2 −
4 4 5 4
Which leaves,
3 5
3
7 = +
5 4
3
Subtract from both sides,
5
3 5
3
7 − =
5 4
Which becomes,
19
1 34 3 5
3
(7 − ) = =
5 5 4
5
Multiply both sides by to give,
34
5 5 25
3
= × = = 0.1838
4 34 136
Therefore,
3
= √0.1838
The keystrokes for this calculation are,
CqS0.1838=
To confirm this is correct, consider the original
expression,
3 3
3
+ 7 = + 2
4 5
The keystrokes to solve this expression are,
C3a4$+7Q(X)q(x )Q(=)J
3
q(x )a5$+2
3
20
q(SOLVE)
(Note your calculator may have a difference number
in the display from a previous calculation).
=
Which confirms the answer.
Example 3-2:
Solve for x in the following equation,
+
3
=
− 2
Where A = 7.169, B = 6.43, C = 8.46 and D = 1.59
21
The expression can be written as,
3 +
= √
− 2
The keystrokes for your fx calculator to perform this
calculation are,
CqSaQ(A)+Q(B)RQ(C)p
Q(D)d
r
A? 7.169=
B? 6.43=
C? 8.48=
D? 1.59=
The value of x is therefore 1.31857
22
Example 3-3:
Solve for x in the following equation,
2
3
= √2.2 −
7
After rearranging you should arrive at,
3 2 2
= √ 2.2 − ( )
7
The keystrokes for this calculation are,
CqS2.2p(2a7$)
d=
To confirm this result, the keystrokes for finding x
using SOLVE are,
C2a7$Q(=)s2.2pQ(X)
q(x )
3
23
q(SOLVE)
1=
The value of x = 1.284, this confirms the result you
should have obtained by re-arranging the expression.
Available from
www.CambridgePaperbacks.com
24
4 Trigonometric Expressions
In this chapter you will encounter trigonometric
functions are their inverse values which you should be
able to derive. Here are two fully worked examples.
Example 4-1:
Solve for x in the following equation,
2
√1 − = sin(27.7 )
Square both sides of the expression,
2
2
1 − = sin (27.7 )
Add -1 to both sides,
2
2
1 − − 1 = sin (27.7 ) − 1
Leaving
2
2
= 1 − sin (27.7 )
Take the square root of both sides,
2
= √1 − sin (27.7 )
The keystrokes for the calculation are,
25
Cs1pj27.7q(Deg)1)
d=
Leaving x = 0.8854. The (Deg) keystroke is found
above the M key; it’s useful to use this should the
calculator be in it’s radian mode. This result can be
confirmed using SOLVE with the following keystrokes,
2
√1 − = sin(27.7 )
Cs1pQ(X)d$Q(=)j27.
7q(Deg)1)
q(SOLVE) 1=
Which confirms the result.
26
Example 4-2:
Solve for x in the following equation for x in the range
o
o
[15 , 25 ]
7
= cos(2 − 1)
9
Take the inverse cosine of both sides,
7
cos −1 ( ) = cos −1 (cos(2 − 1))
9
7
2 − 1 = cos −1 ( )
9
Add 1 to both sides,
7
2 − 1 + 1 = cos −1 ( ) + 1
9
7
2 = cos −1 ( ) + 1
9
Divide both sides by 2,
1 7
= [cos −1 ( ) + 1]
2 9
The keystrokes for this calculation are,
27
C1a2$(q>7a9$)
+1)=
x
o
Therefore x = 19 , 58 minutes and 16.39 seconds. You
will have notice we used the x key to convert an
angle into minutes and seconds. To confirm this result
using SOLVE, enter the following keystrokes,
7
= cos(2 − 1)
9
C7a9$Q(=)k2Q(X)p1)
o
o
We are looking for a solution in the range [15 , 25 ]
q(SOLVE) 15=
Which confirms the result.
28
In the following examples, re-arrange the expressions
yourself to get x on the left hand side and perform the
calculation.
Example 4-3:
Solve for x by re-arranging the following equation
1 − 3
= tan(54.3 )
3.2
After re-arranging you should arrive at the result,
3
= √1 − 3.2 tan(54.3 )
The keystrokes for this calculation are,
CqS1p3.2l54.3
)=
To confirm this result, the keystrokes for finding x
using SOLVE are,
Ca1pQ(X)q(x )R3.2$
3
Q(=)l54.3q(Deg)1)
29
q(SOLVE) 1=
x = -1.5115 which should confirm your result from re-
arranging the expression.
Example 4-4:
Solve for x by re-arranging the following equation
16.8
2
3 − = 0
sin(65.2 )
After re-arranging you should arrive with,
3 16.8
= √
3 sin(65.3)
The keystrokes for this calculation are,
CqS16.8a3j65.
3)=
30
To confirm this result, the keystrokes to solve this
expression are,
C3Q(X)dp16.8aJ(X)j
65.2q(Deg)1)$Q(=)0
q(SOLVE) 1=
The value of x = 1.834
Example 4-5:
Find the value of x in the expression,
1 +
= |cos ( )|
2 2
When
31
1 A = 85.3, B = 102.8 and C = 5.7
2 A = 55.2, B = 15.8 and C = 2.5
After re-arranging your should arrive with,
1
=
+
√2 |cos ( )|
The keystrokes for this calculation are,
C1as2q(Abs)kaQ(A)+
Q(B)RQ(C)$)r
The keystrokes for the original expression are,
C1a2Q(X)d$Q(=)q(Abs)k
aQ(A)+Q(B)RQ(C)$)
You will see the expression in the display of your fx
calculator,
1 q(SOLVE)
A? 85.3=
B? 102.8=
32
C? 5.7=
Solve for X 1=
The value of x is 0.772.
2 q(SOLVE)
A? 55.2=
B? 15.8=
C? 2.5=
Solve for X 1=
The value of x is 0.7109
Example 4-6:
Re-arrange the following expression to find x.
2
+ tan(3 − 1) = 2.6
11.5
33
After re-arranging this expression you should arrive
with,
1 2
= [1 + tan −1 (2.6 − )]
3 11.5
The keystrokes for this calculation are,
C1a3$(1+q?2.6
p2a11.5$))=
To confirm this result, enter the following keystrokes,
C2a11.5$+l3Q(X)p
1)Q(=)2.6q(SOLVE)
Solve for X 1=
Leaving x = 22.866.
34
5 Logarithmic and Exponential Expressions
In this chapter we shall be looking at mix of
logarithmic and exponential expressions as they are
both closely linked.
Example 5-1:
Find the value of x in the following expression.
5.4
= log (3.2 + 2)
3
3
Express this as,
5.4
log (3) = log (3.2 + 2)
3
3
3
This can written as,
5.4
log (3 3 ) = log (3.2 + 2)
3
3
Remove the logs,
5.4
3 3 = 3.2 + 2
Add -2 to both sides,
5.4
3.2 = 3 3 − 2
35
Divide both sides by 3.2
5.4
3 3 − 2
=
3.2
The keystrokes for performing this calculation are,
Ca3^5.4P3$p2R3
.2=
The value of x is 1.6327, to confirm this result use
SOLVE on your fx-991ES calculator with the following
keystrokes,
5.4
= log (3.2 + 2)
3
3
C5.4a3$Q(=)i3$3.
2Q(X)+2
q(SOLVE)
36
Solve for X 1=
Which confirms the result.
Example 5-2:
Re-arrange the following expression to find the value
of x. when A = 5.87, B = 8.419 and C = 2.097
1.7 =
−
Take logs on both sides,
ln( 1.7 ) = ln ( )
−
This becomes,
1.7 = ln ( )
−
Divide both sides by 1.7
1
= ln ( )
1.7 −
37
The keystrokes for this expression are,
C1a1.7$hQ(A)aQ(B)p
Q(C)$)
r
A? 5.87=
B? 8.419=
C? 2.097=
The value of x = -0.0436. To confirm this result, use
SOLVE on the original expression; the keystrokes are,
1.7 =
−
CqH1.72Q(X)$Q(=)
Q(A)aQ(B)pQ(C)
38
q(SOLVE) ===1=
which confirms the result is correct. You should now
be able to calculate the values of x in the following
examples.
Example 5-3:
Find the value of x in the following expression by re-
arranging the expression to make x the subject,
log ( + 2) = 1.2
7
After re-arranging you should arrive with,
= 7 1.2 − 2
The keystrokes for this calculation are,
C7^1.2$p2=
To confirm this solution, enter the following
keystrokes into the fx calculator,
39
Ci7$Q(X)+2$Q(=)1.2
q(SOLVE)
Solve for X 1=
The value of x is 8.3304 which should correspond with
the result you obtained.
Example 5-4:
Find the value of x in the following expression when
1. A = 7.169, B = 5.82 and C = 3.382
2. A = 8.46, B =8.97 and C = 7.17
+
10 1.4 =
After re-arranging you should arrive with,
40
1 +
= log ( )
1.4
The keystrokes for this calculation are,
C1a1.4$gaQ(A)+Q(B)
RQ(C)$)r
To confirm this result, enter the following keystrokes,
CqG1.4Q(X)$Q(=)a
Q(A)+Q(B)RQ(C)
1 q(SOLVE)
A? 7.169=
B? 5.82=
C? 3.382=
Solve for X 1=
The value of x is 0.4174 which should correspond with
the first result you obtained.
41
2 q(SOLVE)
A? 8.46=
B? 8.97=
C? 7.17=
Solve for X 1=
The value of x is 0.2755 which should correspond with
the second result you obtained.
Available from
www.CambridgePaperbacks.com
42
6 Indices
Although you have been using some indices, in this
chapter we shall look at specific examples involving
indices.
Example 6-1:
Find the value of x which satisfies the expression,
1
= 0.488
2 + 2.3
Invert both side of the expression,
1
2 + 2.3 =
0.488
Add -2 to both sides,
1
2 + 2.3 − 2 = − 2
0.488
leaving
1
2.3 = − 2
0.488
2.3 1
= √ − 2
0.488
43
The keystrokes for performing this calculation are,
CF2.3$1a0.488$
p2=
The value of x is 0.2699. To confirm this result, enter
the following keystrokes using the SOLVE feature on
your fx calculator,
1
= 0.488
2 + 2.3
C1a2+Q(X)^2.3$$
Q(=)0.488
q(SOLVE) Solve for X 1=
Which confirms the result.
44
Example 6-2:
Find the value of x which satisfies the expression,
1 − 1.9
= 0.377
1 + 1.9
After re-arranging you should arrive with,
1.9 1 − 0.377
= √
1 + 0.377
The keystrokes for this calculation are (using M),
0.377=
qF1.9$a1pMR1+
M=
Confirm this result with the following keystrokes for
the original expression,
Ca1pQ(X)^1.9R1+
J(X)^1.9$$Q(=)0.377
45
q(SOLVE) Solve for X 1=
Which should confirm the result you obtained.
Example 6-3:
Find the value of x which satisfies the expression,
1
√ = 1.8665
1 − −1.6
After re-arranging you should arrive with,
1 1
= − ln (1 − )
1.6 1.8665 2
The keystrokes for this calculation are,
Cz1a1.6$h1p1a1
.8665d$)=
46
The words you are searching are inside this book. To get more targeted content, please make full-text search by clicking here.
Discover the best professional documents and content resources in AnyFlip Document Base.
Search