algebra 1

Classification of numbers in the set $\ impossibility \{-3, 0, \sqrt{2}, \frac{3}{4}, \pi, 5\}$

Natural: $5$; Whole: $0, 5$; Integer: $-3, 0, 5$; Rational: $-3, 0, \frac{3}{4}, 5$; Irrational: $\sqrt{2}, \pi$; Real: All listed numbers

Definition and properties of the set of Irrational Numbers

Real numbers that cannot be expressed as a fraction $\frac{a}{b}$ of two integers $a$ and $b$ (where $b \neq 0$); their decimal expansions are non-terminating and non-repeating (e.g., $\pi$, $\sqrt{2}$).

Identity Property of Addition vs. Identity Property of Multiplication

Additive Identity: Adding $0$ leaves a number unchanged ($a + 0 = a$). Multiplicative Identity: Multiplying by $1$ leaves a number unchanged ($a \times 1 = a$).

Associative Property of Addition and Multiplication

Changing the grouping of terms or factors does not change the result: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.

Simplify: $12 - 2 \times |3 - 8| + 4^2$

$$18$$

Simplify: $\frac{2^3 + 4 \times 3}{10 - 2 \times 3}$

$$5$$

Evaluate $-x^2 - 3xy$ when $x = -2$ and $y = 4$

$$20$$

Find the error and solve correctly: $3(x - 4) = 15 \rightarrow 3x - 4 = 15 \rightarrow 3x = 19 \rightarrow x = \frac{19}{3}$

Mistake: The $3$ was not distributed to $-4$ in the first step. Correct step: $3x - 12 = 15$ leading to $3x = 27$ and $x = 9$.

Solve for $x$: $-\frac{2}{3}x + 5 = -1$

$$x = 9$$

Solve for $x$: $4(x - 2) = 2(2x + 1) - 10$

Infinitely many solutions (Identity), because simplifying both sides yields $4x - 8 = 4x - 8$.

Solve for $x$: $5x - 3(x + 2) = 4x - 9$

$$x = \frac{3}{2}$$

Rule for flipping the inequality symbol during algebraic manipulation

Reverse (flip) the inequality symbol whenever you multiply or divide both sides of the inequality by a negative number.

Solve for $x$ and describe its number line graph: $-3x + 7 \le 22$

Algebraic solution: $x \ge -5$. Graph description: A closed (filled-in) circle at $-5$ with a shaded arrow extending to the right.