Course Review

Every number in Algebra 1 is a real number, sorted into nested sets:

• Natural: $1, 2, 3, \dots$  Whole: $0, 1, 2, 3, \dots$
• Integers: $\dots, -2, -1, 0, 1, 2, \dots$
• Rational: any $\frac{a}{b}$ with integers $a,b$ and $b \ne 0$; decimals end or repeat.
• Irrational: cannot be written as a fraction; decimals never end and never repeat (e.g. $\sqrt{2},\ \pi$).

Each set sits inside the next: naturals $\subset$ wholes $\subset$ integers $\subset$ rationals.

• P --- Parentheses / grouping (also $|\,\cdot\,|$ and fraction bars)
• E --- Exponents
• MD --- Multiply and Divide, left to right
• AS --- Add and Subtract, left to right

Multiplication and division are equal in rank; do whichever comes first left to right. Same for addition and subtraction.

multicols2

• Commutative: $a+b=b+a$, $\ ab=ba$
• Associative: $(a+b)+c=a+(b+c)$
• Distributive: $a(b+c)=ab+ac$
• Identity: $a+0=a$, $\ a\cdot 1=a$

multicols Like terms share the same variable part (same letters, same exponents). Combine by adding coefficients: $5x+3x=8x$. Note $x^2$ and $x$ are not like terms.

Goal: isolate the variable. Work backwards through PEMDAS.

• Distribute and clear parentheses; clear fractions by multiplying by the LCD.
• Combine like terms on each side.
• Move variable terms to one side, constants to the other (do the same to both sides).
• Divide by the coefficient.

Variables on both sides: collect all variable terms on the side that keeps the coefficient positive.

Literal equation: solve for one letter in terms of the others using the same isolation steps. Example: $A = \tfrac{1}{2}bh \Rightarrow h = \dfrac{2A}{b}$.

How many solutions?

• Variable remains, e.g. $x = 4$ $\Rightarrow$ one solution.
• True statement, e.g. $6 = 6$ $\Rightarrow$ all real numbers (identity).
• False statement, e.g. $3 = 7$ $\Rightarrow$ no solution.

Solve inequalities like equations, with one rule: multiplying or dividing by a negative flips the inequality sign.

Compound: “AND” means both true (overlap, $a < x < b$); “OR” means either true (union).

Absolute value (for $k>0$):

Slope $m$ measures steepness (rise over run):

Positive $m$ rises left-to-right; negative $m$ falls. Horizontal lines have $m=0$; vertical lines have undefined slope.

• Slope-intercept: $y = mx + b$  ($m$ slope, $b$ is $y$-intercept)
• Point-slope: $y - y_1 = m(x - x_1)$
• Standard: $Ax + By = C$  ($A, B, C$ integers, $A \ge 0$)

Intercepts: set $y=0$ for the $x$-intercept, set $x=0$ for the $y$-intercept.

• Parallel lines have equal slopes: $m_1 = m_2$.
• Perpendicular lines have opposite reciprocal slopes: $m_1 \cdot m_2 = -1$, so $m_2 = -\dfrac{1}{m_1}$.

Function notation: $f(x)$ names the output for input $x$. A relation is a function if each input has exactly one output (passes the vertical line test). $f(3)$ means substitute $3$ for $x$.

A solution $(x,y)$ satisfies both equations --- the point where the graphs cross.

• Graphing: plot both lines; the intersection is the solution.
• Substitution: solve one equation for a variable, plug into the other. Best when a variable is already isolated.
• Elimination: add or subtract equations (scale first) to cancel one variable. Best when lined up in standard form.

• One solution: lines cross once (different slopes).
• No solution: parallel lines, same slope, different intercept (variables cancel, false statement).
• Infinitely many: same line (variables cancel, true statement).

Systems of inequalities: graph each, shade; the solution is the overlap of shaded regions.

For any nonzero base: multicols2

• Product: $x^a \cdot x^b = x^{a+b}$
• Quotient: $\dfrac{x^a}{x^b} = x^{a-b}$
• Power of power: $(x^a)^b = x^{ab}$
• Power of product: $(xy)^a = x^a y^a$
• Power of quotient: $\left(\dfrac{x}{y}\right)^a = \dfrac{x^a}{y^a}$
• Zero: $x^0 = 1$

multicols Negative exponent: $x^{-a} = \dfrac{1}{x^a}$ --- move the factor across the fraction bar to make the exponent positive.

A number written as $a \times 10^n$ where $1 \le a < 10$ and $n$ is an integer.

• Large numbers: positive $n$. $\;5{,}200{,}000 = 5.2 \times 10^6$.
• Small numbers: negative $n$. $\;0.00047 = 4.7 \times 10^{-4}$.

Multiply/divide by combining the $a$ values and adding/subtracting the exponents.

A polynomial is a sum of terms $ax^n$ (whole-number exponents). Degree is the highest exponent; name by terms: monomial (1), binomial (2), trinomial (3).

Add/Subtract: combine like terms. To subtract, distribute the minus sign to every term in the second polynomial first.

FOIL for two binomials --- First, Outer, Inner, Last:

Special products (memorize):

• GCF first, always: pull out the greatest common factor. $\;6x^2 + 9x = 3x(2x + 3)$.
• Difference of squares: $a^2 - b^2 = (a+b)(a-b)$.
• Perfect-square trinomial: $a^2 \pm 2ab + b^2 = (a \pm b)^2$.
• Trinomial $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$.
• Leading coefficient $\ne 1$: factor by grouping (AC method).

If a product equals zero, at least one factor is zero:

This turns a factored equation into solutions (the roots).

The graph is a parabola. It opens up if $a > 0$, down if $a < 0$.

• Axis of symmetry / vertex $x$: $\ x = -\dfrac{b}{2a}$.
• Vertex: $\left(-\dfrac{b}{2a},\ f\!\left(-\dfrac{b}{2a}\right)\right)$ --- the minimum or maximum point.
• Vertex form: $y = a(x - h)^2 + k$ with vertex $(h, k)$.

• Factoring: write as a product $=0$, use zero product property.
• Square roots: for $x^2 = k$, then $x = \pm\sqrt{k}$.
• Completing the square: add $\left(\dfrac{b}{2}\right)^2$ to form a perfect square.
• Quadratic formula (always works):

The discriminant is $\;D = b^2 - 4ac$ (under the radical). It counts real solutions:

• $D > 0$: two real solutions.
• $D = 0$: one (repeated) real solution.
• $D < 0$: no real solutions.

A radical is simplified when no perfect-square factor remains under the root.

Example: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

Add/Subtract: only combine like radicals (same radicand): $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$.

Multiply: $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$.

Rationalize a denominator by removing the root from the bottom:

Pythagorean theorem (right triangles, legs $a,b$, hypotenuse $c$):

A rational expression is a fraction of polynomials. Factor top and bottom, then cancel common factors.

Excluded values: any $x$ making an original denominator $0$ is not allowed (here $x \ne -1$ and $x \ne -3$).

• Multiply: factor, cancel, multiply across.
• Divide: multiply by the reciprocal of the second fraction.
• Add/Subtract: rewrite over a common denominator (LCD), combine numerators.
• Solve rational equations: multiply every term by the LCD to clear fractions, solve, then discard any answer that is an excluded value.

• Mean (average): $\ \dfrac{\text{sum of values}}{\text{number of values}}$.
• Median: middle value of the ordered list (average the two middle values if the count is even).
• Mode: the value(s) that appear most often.

The median resists outliers (extreme values); the mean gets pulled toward them.

• Range $=$ maximum $-$ minimum.
• Quartiles: $Q_1$ (median of lower half), $Q_2$ (median), $Q_3$ (median of upper half).
• Interquartile range: $\ \text{IQR} = Q_3 - Q_1$ --- the spread of the middle $50\%$.

A scatter plot shows paired data to reveal a relationship (correlation):

• Positive: both rise together.  Negative: one rises as the other falls.
• No correlation: no clear pattern.

The line of best fit models the trend; use its equation $y = mx + b$ to predict values. Correlation does not prove causation.