WHS Algebra Final Exam Review - Module 1

Module 1 Review: Relationships Between Quantities and Reasoning with Equations and Their Graphs

• This review covers Module 1 (Lessons 1 to 24) of the WHS Algebra curriculum.
• General instructions indicate that notebooks are permitted for use during the final exam review.

Fundamental Concepts and Multiple Choice

• Standard Form of Polynomials:

  • A polynomial is simplified and written in standard form when its terms are ordered by the largest power (exponent) first, following a decreasing (declension) order of powers.
  • To reach standard form, one must first combine like terms and apply the distributive property where necessary.
  • Example: In the polynomial $2x^3 + 7x + 20$, the powers are $3, 1, \text{and } 0$, which is in proper standard form.

• Equating Expressions and Variables:

  • Given the equation $(x + a)(x - 5) = x^2 + bx - 10$, we can find the values of $a$ and $b$ by expanding the left side via the FOIL method or distributive property:
    • $(x)(x) + (x)(-5) + (a)(x) + (a)(-5) = x^2 + (a - 5)x - 5a$
  • By matching the constant terms: $-5a = -10 \Rightarrow a = 2$.
  • By matching the x-coefficients: $b = a - 5$. Substituting $a = 2$ gives $b = 2 - 5 = -3$.
  • Final Answer: $a = 2, b = -3$.

• Algebraic Properties:

  • Commutative Property of Addition: States that the order of addition does not change the sum. Equation format: $A + B = B + A$.
  • Example display: $a(b + x) + b = b + a(b + x)$.

• Solution Sets for Inequalities:

  • For the sentence $2x \geq x - 12$, solving for $x$ involves subtracting $x$ from both sides: $x \geq -12$.
  • Any value less than $-12$ is NOT part of the solution set.
  • For the compound inequality $t < -1 \text{ or } t > 5$:
    • Solution includes values like $-5$ (since $-5 < -1$) and $10$ (since $10 > 5$).
    • The values $-1$ and $5$ are NOT part of the solution set because the inequalities are strict (no "equal to" component).

• Zero Product Property:

  • For the equation $(r - 2)(3r + 5)(r + 4) = 0$, the solutions (roots) are found by setting each factor to zero:
    • $r - 2 = 0 \Rightarrow r = 2$
    • $3r + 5 = 0 \Rightarrow 3r = -5 \Rightarrow r = -\frac{5}{3}$
    • $r + 4 = 0 \Rightarrow r = -4$
  • Solution set: $\{2, -\frac{5}{3}, -4\}$.

• Equation Equivalence:

  • Two equations have the same solution set if one can be transformed into the other via algebraic operations.
  • For an equation such as $\frac{3}{5}x = 6$, multiplying both sides by $5$ results in $3x = 30$.

• Testing Ordered Pairs in Inequalities:

  • To determine if a point $(x, y)$ is a solution to $5x - 2y \geq 3$, substitute the values into the variables.
  • Example: For $(-2, 1)$, $5(-2) - 2(1) = -10 - 2 = -12$. Since $-12$ is not $\geq 3$, the pair $(-2, 1)$ is NOT a solution.

• Linear Equation Forms:

  • Slope-Intercept Form: Written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
  • Converting $3x + 5y = 25$:
    1. Subtract $3x$ from both sides: $5y = -3x + 25$.
    2. Divide every term by $5$: $y = -\frac{3}{5}x + 5$.

Operations and Simplifying Expressions

• Polynomial Multiplication:

  • $(x - 6)(x + 5)$: Using the distribution method (often referred to as the XBOX or grid method in student notes):
    • $x^2 + 5x - 6x - 30 = x^2 - x - 30$
  • $(x - 7)(x + 7)$: This is a difference of squares pattern.
    • $x^2 + 7x - 7x - 49 = x^2 - 49$
  • $(x^2 - 2)(x^2 - 9x + 1)$: Distribute each term in the first binomial to each in the second:
    • $x^2(x^2 - 9x + 1) - 2(x^2 - 9x + 1)$
    • $x^4 - 9x^3 + x^2 - 2x^2 + 18x - 2$
    • Simplified: $x^4 - 9x^3 - x^2 + 18x - 2$
  • Square of a Binomial: $(x - 6)^2 = (x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36$.

• Polynomial Subtraction:

  • $(6v - 3) - (11v^3 - 2v + 25)$: Distribute the negative sign to the second polynomial.
    • $6v - 3 - 11v^3 + 2v - 25$
    • Grouping like terms: $-11v^3 + (6v + 2v) + (-3 - 25)$
    • Simplified: $-11v^3 + 8v - 28$

• Exponents and Powers:

  • Power of a Product with Distribution: $3t^2s(3t^2s^4 + 4s - t^3)$
    • $(3t^2s \cdot 3t^2s^4) + (3t^2s \cdot 4s) - (3t^2s \cdot t^3)$
    • Simplified: $9t^4s^5 + 12t^2s^2 - 3t^5s$
  • Power of a Power Rule: $(x^2x^6)^3 = (x^8)^3 = x^{8 \times 3} = x^{24}$.
  • Quotient Rule: $\frac{p^8}{p^3} = p^{8-3} = p^5$.
  • Simplifying Monomial Fractions: $\frac{12j^5}{4j^{15}} = \frac{12}{4} \cdot j^{5-15} = 3j^{-10} = \frac{3}{j^{10}}$.
  • Negative Exponents: $(5)^{-3} = \frac{1}{5^3} = \frac{1}{125}$.

Solving Equations and Inequality Representations

• Solving and Graphing on Number Lines:

  • Equation: $16a - 10 = 22$
    • $16a = 32 \Rightarrow a = 2$.
    • Set notation: $\{2\}$.
    • Graphical: A solid point at 2.
  • Variables on Both Sides: $7x - 4(x + 1) = 6 + 2x$
    • Distribute: $7x - 4x - 4 = 6 + 2x$
    • Combine: $3x - 4 = 6 + 2x$
    • Solve: $x = 10$.
    • Set notation: $\{10\}$.
  • Inequality: $4y > -30 + y$
    • $3y > -30 \Rightarrow y > -10$.
    • Graphical: Open circle at $-10$, shaded to the right.
  • Multi-Step Inequality: $5w \leq 25$
    • $w \leq 5$.
    • Graphical: Closed circle at 5, shaded to the left.

• Compound Sentences:

  • OR (Union): $p < 2 \text{ or } p > 4$.
    • Graphical: Shading extends outward from open circles at 2 and 4.
  • Inequality Simplification: $c + 5 \leq 2 \text{ or } 3c \leq 5c - 6$
    • $c \leq -3 \text{ or } -2c \leq -6 \Rightarrow c \geq 3$.
    • Graphical: Solid circles at $-3$ (shaded left) and $3$ (shaded right).
  • AND (Intersection): $-4 < 5 + 9x \leq 23$
    • Subtract 5: $-9 < 9x \leq 18$
    • Divide by 9: $-1 < x \leq 2$.
    • Graphical: Open circle at $-1$, closed circle at 2, shaded between them.
  • Contradiction/Agreement: $m \geq 0 \text{ and } m \geq 2$
    • Since both must be true, the overlapping region is $m \geq 2$.

• Sequence of Properties in Equation Solving:

  • For the equation $9(2z - 3) = -7 + 13z$:
    • Step 1: $18z - 27 = -7 + 13z$ (Distributive Property).
    • Step 2: $18z = 20 + 13z$ (Addition Property of Equality / Inverse Operation).
    • Step 3: $5z = 20$ (Subtraction Property of Equality).
    • Step 4: $z = 4$ (Division Property of Equality).

Advanced Solving and System Analysis

• Rational Equations:

  • Method: Cross Products.
  • Case 1: $\frac{3x}{x - 2} = \frac{5}{2}$
    • $3x(2) = 5(x - 2) \Rightarrow 6x = 5x - 10 \Rightarrow x = -10$.
    • Constraint: $x \neq 2$ (Value cannot be 2 as it makes the denominator zero).
  • Case 2: $\frac{5x - 5}{x - 1} = 5$
    • $5(x - 1) = 5(x - 1)$. This is an identity.
    • Constraint: $x \neq 1$.
    • Solution set: All real numbers except 1.

• Literal Equations (Solving for Formulas):

  • Solve $F = \frac{G \cdot m}{d^2}$ for $m$.
  • Multiply by denominator: $F \cdot d^2 = G \cdot m$.
  • Isolate $m$: $m = \frac{F \cdot d^2}{G}$.

• Systems of Linear Equations:

  • Elimination Method Strategy:
    • Given: $5x - 2y = 3$ and $2x + 7y = 9$.
    • To eliminate $x$, the coefficients must be opposite numbers.
    • Strategy: Multiply the first equation by $2$ and the second by $-5$ (making them $10x$ and $-10x$), then add the equations.
  • Substitution Method:
    • If $y = 4x + 1$ and $4x - 2y = 6$:
    • Substitute the expression for $y$ into the second equation: $4x - 2(4x + 1) = 6$.
    • Solve: $4x - 8x - 2 = 6 \Rightarrow -4x = 8 \Rightarrow x = -2$.
    • Substitute back to find $y$: $y = 4(-2) + 1 = -7$.
    • Solution: $(-2, -7)$.
  • Elimination Example:
    • $9a + b = 18$
    • $3a - b = -6$
    • Add equations: $12a = 12 \Rightarrow a = 1$.
    • Find $b$: $9(1) + b = 18 \Rightarrow b = 9$.
    • Solution: $(1, 9)$.
  • Graphing Method:
    • To solve graphically, identify the intersection point of the lines.
    • Example Intersection: $(0, -3)$ or as indicated by coordinates on the grid.

• Graphing Inequalities:

  • For the system of inequalities $y = -x + 4$ (as a boundary) and $y < x - 3$, use dashed lines for strict inequalities ($<$, $>$) and solid lines for inclusions ($\leq$, $\geq$).
  • Shading occurs based on the truth value of a test point (usually $(0,0)$).