≡Operations with Rational Numbers
What are rational numbers?
They are numbers that can be written in fraction form:
\( \frac{a}{b}, \quad b \neq 0 \)
Examples:
\( \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} \)
valor absoluto
The valor absoluto represents the distance to zero:
\( |a| = a \) se \( a \geq 0 \)
\( |a| = -a \) se \( a < 0 \)
Examples:
\( |5| = 5 \)
\( |-5| = 5 \)
Simétrico
The Simétrico of a number is:
\( -a \)
Examples:
\( 3 \rightarrow -3 \)
\( -7 \rightarrow 7 \)
Adição e subtração
Addition with the same sign
Add the values and keep the sign
Examples:
\( 3 + 5 = 8 \)
\( -4 + (-6) = -10 \)
Soma com sinais contrários
Subtract the values and take the sign of the number with the greatest absolute valueNL# Examples:
\( 7 + (-3) = 4 \)
\( -8 + 5 = -3 \)
Soma de números simétricos
The result is always zero
\( a + (-a) = 0 \)
Examples:
\( 5 + (-5) = 0 \)
Subtração (diferença)
It is transformed into the sum of the opposite
\( a - b = a + (-b) \)
Examples:
\( 7 - 3 = 7 + (-3) = 4 \)
\( 5 - (-2) = 5 + 2 = 7 \)
Divisão de números racionais
Main rule
Multiply by the reciprocal
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \) Inverso de um número racional
Swap the numerator and denominator
\( \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 \)
Examples:
\( \frac{3}{4} \rightarrow \frac{4}{3} \)
\( 2 = \frac{2}{1} \rightarrow \frac{1}{2} \)
Important:
\( 0 \text{ não tem inverso} \)
Examples de divisão
\( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)
TIPS
✔ Same signs → add and keep the sign ✔ Different signs → subtract and take the sign of the number with the greatest absolute value ✔ Subtraction → convert to addition ✔ Division → multiply by the reciprocal ✔ Número + simétrico: \( a + (-a) = 0 \)
✔ Valor absoluto: \( |a| \geq 0 \)
Real Numbers & Powers
1. Powers with integer exponents
- General rule:
\( a^n \) means multiplying 𝑎 by itself 𝑛 times
✔ Example:
\( 2^3 = 2 \times 2 \times 2 = 8 \)
- Zero exponent:
\( a^0 = 1 \), with \( a \neq 0 \)
✔ Example:
\( 5^0 = 1 \)
- Negative exponent:
\( a^{-n} = \frac{1}{a^n} \), with \( a \neq 0 \)
✔ Example:
\( 2^{-3} = \frac{1}{8} \)
- Rules important:
\( a^m \times a^n = a^{m+n} \)
\( \frac{a^m}{a^n} = a^{m-n} \)
\( (a^m)^n = a^{m \cdot n} \)
✔ Example:
\( 2^3 \times 2^2 = 2^5 = 32 \)
2. Scientific Notation
- Form:
\( a \times 10^n \), with \( 1 \leq a < 10 \)
✔ Examples
\( 3000 = 3 \times 10^3 \)
\( 0.004 = 4 \times 10^{-3} \)
3. Real Numbers and Decimals
- Real Numbers
Include: integers, rational and irrational numbers
- Decimals
✔ Finite decimal:
\( 0.5 = 1/2 \)
✔ Repeating decimal:
\( 0.333... = 1/3 \)
4. Operations with Real Numbers
- Order of operations:
Parentheses
Powers
Multiplication and division
Addition and subtraction
✔ Example:
\( 2 + 3 \times 4 = 2 + 12 = 14 \)
Test yourself in these challenges 👇
Discover some interesting facts about Mathematics EN 9th grade
Order, Intervals and Approximations
1. Order Relations in 𝑅
- Symbols
\( > \), \( < \), \( \geq \), \( \leq \)
✔ Example:
\( -3 < 2 \)
2. Approximate Values:
- Rounding
✔ Example:
\( 3{,}46 \approx 3{,}5 \)
- Truncation
✔ Example:
\( 3{,}46 \approx 3{,}4 \)
3. Intervals of real numbers
- Types:
\( [a, b] \)\( [a, b] \) (includes endpoints)
\( ]\( ]a, b[ \)a, b[ \) (does not include endpoints)
✔ Examples:
\( [1, 3] \)
\( ]1, 3[ \)
4. Intersection of intervals:
👉 Common part
✔ Example:
\( [1, 5] \cap [3, 7] = [3, 5] \)
5. Interval Union
👉 Union of intervals
✔ Example:
\( [1, 3] \cup [2, 5] = [1, 5] \)
QUICK TIPS FOR THE EXAM
✔ Negative exponent → becomes a fraction
✔ Scientific notation → number between 1 and 10
✔ Order → pay attention to negative signs
✔ Intervals → brackets include, parentheses do not include
✔ Intersection → common part
✔ Union → everything together
Introduction to Polynomials
Monomials and Polynomials
- Monomials
Expression with a single term
\( 3x^2,\ -5x,\ 7 \)
- Polynomial
Sum of monomials
\( 3x^2 + 2x - 5 \)
2. Sum of polynomials
Like terms are added
✔ Example:
\( (2x + 3) + (x + 5) = 3x + 8 \)
3. Product of a monomial by a polynomial
Distribute the monomial
✔ Example:
\( 2x(3x + 4) = 6x^2 + 8x \)
4. Operations with polynomials
✔ Example:
\( (x + 2)(x + 3) = x^2 + 5x + 6 \)
5. Notable Cases
- Square of a binomial
\( (a + b)^2 = a^2 + 2ab + b^2 \)
✔ Exemplo:
\( (x + 2)^2 = x^2 + 4x + 4 \)
- Diferença de quadrados
\( a^2 - b^2 = (a - b)(a + b) \)
✔ Example:
\( x^2 - 9 = (x - 3)(x + 3) \)
6. Polynomial Factorization
- Distributive Property
\( ax + ay = a(x + y) \)
✔ Example:
\( 3x + 6 = 3(x + 2) \)
- Difference of Squares
\( x^2 - 16 = (x - 4)(x + 4) \)
- Binomial Square
✔ Example:
\( x^2 + 4x + 4 = (x + 2)^2 \)
First and second degree equations
1. First-degree equations
Form:
\( ax + b = 0 \)
✔ Example:
\( 2x + 4 = 0 \Rightarrow x = -2 \)
2. First-degree inequalities
✔ Example:
\( 2x + 1 > 5 \Rightarrow x > 2 \)
3. Quadratic Equations
- General Form
\( ax^2 + bx + c = 0 \), com \( a \neq 0 \)
- Zero-Product Law
\( ab = 0 \Rightarrow a = 0 \ \text{ou} \ b = 0 \)
- Incomplete Equations
✔ Example:
\( x^2 - 9 = 0 \Rightarrow x = \pm 3 \)
- Complete Equations
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
✔ Example:
\( x^2 - 5x + 6 = 0 \Rightarrow x = 2 \ \text{ou} \ x = 3 \)
4. Literal Equations
Solving for a function of one variable
✔ Example:
\( ax + b = 0 \Rightarrow x = -\frac{b}{a} \)
5. Systems of Equations
✔ Example:
\(
\begin{cases}
x + y = 5 \\
x - y = 1
\end{cases}
\Rightarrow x = 3,\ y = 2
\)
- Classification:
Possible and determined system (one solution)
Impossible system (no solution)
Indeterminate system (infinite solutions)
QUICK TIPS FOR THE EXAM/b>
✔ Similar terms → same literal part
✔ Always distribute correctly
✔ Notable cases → memorize formulas
✔ Zero product → separate into two equations
✔ Always check the solutions
translate this: 1. Definição de função
1. Definition of a function
A function is a relation that associates each value of x with a unique value of y.
\( y = f(x) \)
👉 For each x, there is only one y.
✔ Example:
\( f(x) = 2x + 1 \)
If 𝑥 = 2, then
\( f(2) = 2 \cdot 2 + 1 = 5 \)
2. Domain of a function
It is the set of values of 𝑥 that can be used
\( D_f \)
✔ Example:
\( f(x) = \frac{1}{x} \Rightarrow x \neq 0 \)
👉 Domain: all real numbers except zero
3. Codomain (range set)
It is the set of all possible output values
\( \mathbb{R} \)
👉 Usually these are the real numbers
4. Image (values that the function assumes)
These are the values that actually appear as a result
✔ Example:
\( f(x) = x^2 \)
👉 The image consists only of positive values or zero
5. Independent and dependent variables
- Independent variable → chosen value
\( x \)
- Dependent variable → depends on x
\( y = f(x) \)
✔ Example:
\( y = 3x \)
👉 If x changes, y changes
6. Graphical Representation
It is the graph of the function on the Cartesian plane. Each point has the form:
\( (x, f(x)) \)
✔ Example:
\( f(x) = x + 1 \)
Points on the graph:
\( (0,1), (1,2), (2,3) \)
👉 Forms a straight line
QUICK TIPS FOR THE EXAM
✔ Function → each x has a unique y
✔ Domain → possible values of x
✔ Range → obtained values
✔ Graph → points (x, f(x))
✔ x independent, y dependent
Linear function
1. Definition of an Affine Function
An affine function is a function of the type:
\( f(x) = mx + b \)
👉 where:
𝑚 → slope (inclination of the line)
𝑏 → ordinate at the origin
✔ Example:
\( f(x) = 2x + 1 \)
2. Graphical Representation
👉 The graph of an affine function is always a straight line.
Each point has the form:
\( (x, f(x)) \)
✔ Example:
\( f(x) = x + 1 \)
Points:
\( (0,1), (1,2), (2,3) \)
3. Vertical Line
A vertical line has the equation:
\( x = k \)
👉 It does not represent a function, because one value of x has several values of y.
4. Slope and y-intercept
- Slope (m)
Indicates the inclination of the line
𝑚 > 0 → increasing line
𝑚 < 0→ decreasing line
𝑚 = 0 → horizontal line
- ordinate at the origin (b)
It is the value of 𝑦 when:
\( x = 0 \)
✔ Example:
\( f(x) = 2x + 3 \Rightarrow b = 3 \)
5. Determining the slope
The slope between two points is:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
✔ Example:
\( (1,2) \ \text{e} \ (3,6) \)
\( m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \)
6. Equation of a line
General form:
\( y = mx + b \)
✔ Example:
\( y = 3x - 2 \)
7. Equation of a line (given a point and the slope)
Formula:
\( y - y_1 = m(x - x_1) \)
✔ Example:
Point (1,2) and slope 𝑚=3
\( y - 2 = 3(x - 1) \)
\( y = 3x - 1 \)
QUICK TIPS FOR THE EXAM
✔ Linear function → straight line:
\( f(x) = mx + b \)
✔ 𝑚 → slope
✔ 𝑏 → where the line intersects the 𝑦 axis
✔ Slope → use two points
✔ Vertical line → not a function
✔ Equation → know how to convert from point + slope
Test yourself in these challenges 👇
HOME