Sec 1 Maths Notes Equations
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1) The document provides steps for solving simple linear equations with no fractions and fractional equations. 2) For linear equations with no fractions, the steps are to expand if there is a bracket, group like terms to one side, and then solve for the variable. 3) For fractional equations, the steps are to multiply both sides by the common denominator to clear the fractions, then group like terms and solve.
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![4048 1EM Algebra - Solving (1) Math Academy®
© All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or
stored in any retrieval system of any nature without prior permission.
www.MathAcademy.sg 1
Write your injuries in dust, your benefits in marble
Benjamin Franklin quote
Notes 3: Solving Simple Equations
[A] Solving Linear Equations with no fractions
Step 1:
Step 2:
Example 1: Solve
Example 2: Solve .
532195 +=- xx
532195 +=- xx
5 19 2 53x x- = +
3 19 53x - =
3 19 53x - =
3 72x =
24x =
46)4(2 -=+ xx
46)4(2 -=+ xx
2 8 6 4x x+ = -
2 6 4 8x x- = - -
4 12x- = -
12
4
x
-
=
-
3x =
Expand (if bracket present)
Group like terms to one side
Expand
Group like terms
2x- 2x-
+19 +19
Balance LHS and RHS
Negative 4, not +
concept
faster](https://image.slidesharecdn.com/1em1notesequations-200212033108/85/Sec-1-Maths-Notes-Equations-1-320.jpg)
![4048 1EM Algebra - Solving (1) Math Academy®
© All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or
stored in any retrieval system of any nature without prior permission.
www.MathAcademy.sg 2
[B] Solving Fractional Equations
Method Method Method
Example 6: Solve the equation [3]
Example 7: Solve Ans:
2
2
43
-=
+
x
x
2
2
43
-=
+
x
x
3 4 2 4x x+ = -
3 2 4 4x x- = - -
8x = -
6
5
4
32 -
=
+ rr
4
3
4-
Multiply by denominator L and R
1 2
Cross - Multiply
2´2´
Multiply by Denominator L and R1
3 4 2
2 1
x x+ -
=
(1)(3 4) 2( 2)x x+ = -
3 4 2 4x x+ = -
3 2 4 4x x- = - -
8x = -
3
Make same denominator
Cross - multiply2
Make same Denominator3
2 terms](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Here are the steps to solve this equation by factorising:
1) Factorise the left hand side: 3(x - 1)(x + 2)
2) Set each factor equal to 0:
x - 1 = 0
x + 2 = 0
3) Solve for x:
x = 1
x = -2
Therefore, the solutions are x = 1 or x = -2.
Rational expressions ppt
This document discusses simplifying rational expressions by dividing out common factors, factoring numerators and denominators, and dividing common factors. It provides examples of simplifying various rational expressions step-by-step and explains how to identify excluded values that would make denominators equal to zero.
Lecture 03 special products and factoring
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
Sec 3 A Maths Notes Indices
1. The document discusses solving exponential equations with one, two, or three terms using properties of exponents such as changing bases to the same term and equating powers.
2. Examples are provided for solving two-term exponential equations by making the bases equal and equations with three terms by substituting variables, changing bases to the same term, and equating powers.
3. Solving exponential equations as products using properties such as treating exponents as multipliers is also demonstrated through examples.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
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Presentación Relacionada a Suma, Resta y Valor numérico de Expresiones algebraicas.
Multiplicación y División de Expresiones algebraicas.
Productos Notables de Expresiones algebraicas.
Factorización por Productos Notables.
March 26
This document provides steps and examples for factoring trinomials of the form ax^2 + bx + c. It begins with reviewing special cases like difference of squares and perfect square trinomials. It then outlines a 6-step process for factoring general trinomials: 1) multiply the leading coefficient and constant, 2) find factors of their product that sum to the middle term, 3) rewrite replacing the middle term, 4) factor by grouping, 5) factor out common binomials, 6) check for complete factorization. Examples are provided applying these steps to fully factor trinomials.
factoring and the other ones polynomials2.ppt
The document discusses various methods for factoring polynomials:
1) Factoring out the greatest common factor (GCF) of each term.
2) Using the difference of squares formula a2 - b2 = (a + b)(a - b).
3) Grouping terms to produce common binomial factors.
4) Reversing the FOIL process to factor trinomials of the form x2 + bx + c.
5) Factoring polynomials completely by factoring out the GCF and then factoring the remaining terms.
Kyle Taitt CSU WebworkMATH 160 WeBWorK assignment M160-801-F.docx
Kyle Taitt CSU Webwork
MATH 160 WeBWorK assignment M160-801-FA-1.3 due 09/07/2016 at 11:59pm MDT
1. (1 point) Evaluate the following limits:
1. lim
x→∞
5x3 −5x2 −5x
6−4x−2x3
=
2. lim
x→−∞
5x3 −5x2 −5x
6−4x−2x3
=
Answer(s) submitted:
•
•
(incorrect)
2. (1 point) Evaluate
lim
x→∞
−4x6 + 7x3 −5.
Limit =
Answer(s) submitted:
•
(incorrect)
3. (1 point)
A function is said to have a horizontal asymptote if either the
limit at infinity exists or the limit at negative infinity exists.
Show that each of the following functions has a horizontal
asymptote by calculating the given limit.
lim
x→∞
−13x
12 + 2x
=
lim
x→−∞
12x−9
x3 + 2x−15
=
lim
x→∞
x2 −14x−7
4−6x2
=
lim
x→∞
√
x2 + 15x
4−7x
=
lim
x→−∞
√
x2 + 15x
4−7x
=
Answer(s) submitted:
•
•
•
•
•
(incorrect)
4. (1 point) Evaluate
lim
x→∞
√
x4 + 3x3 −6
10x2 + 6
Answer(s) submitted:
•
(incorrect)
5. (1 point) Find the horizontal limit(s) of the following func-
tion:
f (x) =
6x3 −3x2 −3x
8−6x−4x3
and
Answer(s) submitted:
•
•
(incorrect)
6. (1 point) Evaluate the limit
lim
x→∞
(6−x)(8 + 3x)
(3−9x)(2 + 8x)
Answer(s) submitted:
•
(incorrect)
7. (1 point)
Evaluate the following limits. If needed, enter ’INF’ for ∞
and ’-INF for −∞.
(a)
lim
x→∞
√
6 + 5x2
4 + 11x
=
(b)
lim
x→−∞
√
6 + 5x2
4 + 11x
=
Answer(s) submitted:
•
•
(incorrect)
8. (1 point)
Evaluate the following limit. If the answer is positive infinite,
type ”I”; if negative infinite, type ”N”; and if it does not exist,
type ”D”.
lim
x→∞
x + x3 + x5
1−x2 + x4
Answer(s) submitted:
•
(incorrect)
1
9. (1 point) Evaluate the following limit.
lim
x→∞
x4 −4x2 + 2
x5 + 3x3
Answer:
Answer(s) submitted:
•
(incorrect)
10. (1 point) Evaluate the following limits:
1. lim
x→5
2
(x−5)6
=
2. lim
x→−7−
1
x2(x + 7)
=
3. lim
x→3−
2
x−3
=
4. lim
x→5−
2
(x−5)3
=
Answer(s) submitted:
•
•
•
•
(incorrect)
11. (1 point) This problem has two parts. The second part
will appear after you answer the first part correctly.
(a) What are the vertical asymptotes of f (x) =
4x2
x2 −49
? Your
answer should be a number, a list of numbers separated by com-
mas, or None .
Vertical asymptotes at x =
Answer(s) submitted:
•
(incorrect)
12. (1 point) Find the following limit.
lim
x→−8−
x + 3
x + 8
Limit: help (limits)
Answer(s) submitted:
•
(incorrect)
13. (1 point) Evaluate lim
x→0
x−3
x2(x−2)
Limit = help (limits)
Answer(s) submitted:
•
(incorrect)
14. (1 point)
Find the horizontal and vertical asymptotes of each curve.
List them in increasing order. If there is no such asymptote,
enter ”N”.
y =
x3
x2 + 3x−10
x =
x =
y =
Answer(s) submitted:
•
•
•
(incorrect)
15. (1 point) Let f (x) be a function such that
lim
x→∞
f (x) = ∞ lim
x→−∞
f (x) = 7
lim
x→7+
f (x) = ∞ lim
x→7−
f (x) =−∞
Determine the horizontal asymptote.
y =
Determine the vertical asymptote.
x =
Answer(s) submitted:
•
•
(incorrect)
16. (1 point) Let
f (x) =
x2 + 3x−10
3x2 + 13x−10
.
Find the horizontal ...
0.3 Factoring Polynomials
This document provides an overview of various techniques for factoring polynomials, including:
1) Factoring out the greatest common factor (GCF);
2) Factoring by grouping terms with a common factor;
3) Factoring perfect square trinomials where the first and last terms are perfect squares;
4) Factoring trinomials using techniques like the reverse box method or grouping.
Similar to Sec 1 Maths Notes Equations (20)
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Sec 1 Maths Notes Equations
- 1. 4048 1EM Algebra - Solving (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. www.MathAcademy.sg 1 Write your injuries in dust, your benefits in marble Benjamin Franklin quote Notes 3: Solving Simple Equations [A] Solving Linear Equations with no fractions Step 1: Step 2: Example 1: Solve Example 2: Solve . 532195 +=- xx 532195 +=- xx 5 19 2 53x x- = + 3 19 53x - = 3 19 53x - = 3 72x = 24x = 46)4(2 -=+ xx 46)4(2 -=+ xx 2 8 6 4x x+ = - 2 6 4 8x x- = - - 4 12x- = - 12 4 x - = - 3x = Expand (if bracket present) Group like terms to one side Expand Group like terms 2x- 2x- +19 +19 Balance LHS and RHS Negative 4, not + concept faster
- 2. 4048 1EM Algebra - Solving (1) Math Academy® © All rights reserved. No part of this document may be reproduced or transmitted in any form or by any means, or stored in any retrieval system of any nature without prior permission. www.MathAcademy.sg 2 [B] Solving Fractional Equations Method Method Method Example 6: Solve the equation [3] Example 7: Solve Ans: 2 2 43 -= + x x 2 2 43 -= + x x 3 4 2 4x x+ = - 3 2 4 4x x- = - - 8x = - 6 5 4 32 - = + rr 4 3 4- Multiply by denominator L and R 1 2 Cross - Multiply 2´2´ Multiply by Denominator L and R1 3 4 2 2 1 x x+ - = (1)(3 4) 2( 2)x x+ = - 3 4 2 4x x+ = - 3 2 4 4x x- = - - 8x = - 3 Make same denominator Cross - multiply2 Make same Denominator3 2 terms