Exam Prep Recap: Rational & Radical Equations, Intro to Graphing

Course Administration and Exam Preparation

• Module Progress and Grades:

  • Module 3 (Student Rights) is an essential part of the course, often taking $10-14$ hours to complete.

  • Progress on modules outside of class time has been checked, with some students still needing to catch up.

  • Modules (blue part) and knowledge checks (green part) together account for $35\%$ (approximately one-third) of the overall course grade.

  • The remaining grade comes from exams.

  • It is crucial to work concurrently on module (blue) and knowledge check (green) parts as they are closely related despite differing question counts (e.g., $24$ questions for Module 3 blue, $25$ for Module 3 green).

• First Exam:

  • The first exam is scheduled for two weeks from today (from the lecture's date, which was the $21^{st}$).

  • Students can discuss taking the exam earlier individually.

  • There are no other modules due this current week.

• Knowledge Checks:

  • Knowledge checks can be reset to restore the ALEKS Pie if results indicate mastered topics or areas for review.

  • A knowledge check on Module 4 content will be available next Tuesday.

• ALEKS Access and Resources:

  • Solution sheets (blank and filled) for modules are available on Brightspace under "Learning Content" -> "Module Lesson Pages."

  • Direct ALEKS access is at the very bottom of the Brightspace page.

• Practice Test:

  • A practice test with approximately $40$ questions will be available after scheduled Knowledge Check 2.

  • Paper will be provided for working out solutions (not for submission for credit, but for personal use).

  • Students are allowed one full page of reference notes during the exam.

  • Calculators are permitted and required, but not provided by the instructor; graphics calculators are highly recommended if not explicitly required by the course material.

  • Exams will likely require a lockdown browser and be password-protected.

Key Mathematical Concepts: Rational and Radical Equations

• Overview of Difficult Concepts:

  • Rational equations are generally considered more challenging than radical equations due to the reliance on common denominators and prevalent application problems.

  • Radical equations involve eliminating square roots or other roots.

• Rational Equations:

  • Goal: Eliminate denominators.

  • Common Denominator Method:

    • Find the lowest common denominator (LCD) which is often a combination of all unique parts in the denominators.

    • Multiply all terms (including whole numbers, treated as fractions over $1$) by the LCD to clear denominators.

    • Example: For an expression with denominators $4$ and $(b+8)$, the LCD would be $4(b+8)$.

    • Extraneous Solutions: Always check for values that make the original denominator zero ($b \ne -8$ in the example) as these are invalid solutions.

  • Cross-Multiplication Method:

    • Applicable only when there is exactly one fraction (or rational expression) on each side of the equation.

    • It achieves the same result as multiplying by the common denominator but can be quicker for this specific scenario.

    • If there are more than two terms (e.g., a fraction plus a whole number), combine terms or move them to isolate single fractions on each side before cross-multiplying.

  • Solving Steps:

    • After clearing denominators, solve the resulting linear or quadratic equation.

    • For quadratic equations ($b^2$ terms), use factoring or the quadratic formula.

  • Application Problems:

    • Often involve scenarios like "two computers doing a job together."

    • Example: If two computers together do a job in $4$ minutes, they complete $1/4^{th}$ of the job per minute. If one takes $14$ minutes, rate is $1/14$ per minute. If the faster one takes $x$ minutes, rate is $1/x$ per minute. The equation would be $1/4 = 1/14 + 1/x$. The LCD would be $28x$.

    • Watch for extraneous solutions where $x=0$ or any other value makes a denominator zero.

  • Equations with Multiple Letters (Literal Equations):

    • Treat other letters as constants/numbers and isolate the target variable.

    • Example: For $E = IR$, to solve for $I$, you divide by $R$, resulting in $I = E/R$.

    • Common Error in Inputting Answers: When a solution involves multiple terms, ensure correct fraction grouping. For example, if the solution is $(A/(KB)) - N$, it should be entered as such, not $(A-N)/(KB)$, as $N$ does not share the $(KB)$ denominator.

    • Similarly, for the quadratic formula, the entire numerator ($-b \pm \sqrt{b^2-4ac}$) should be divided by $2a$, not just the square root part.

• Radical Equations:

  • Goal: Get the radical term by itself on one side of the equation.

  • Eliminating the Radical: Raise both sides of the equation to the power corresponding to the root (e.g., square both sides for a square root, cube for a cube root).

  • Algebraic Caution: If there are multiple terms on the non-radical side (e.g., $u+1$), remember to expand (FOIL) them when squaring or cubing.

  • Checking Solutions: It's often possible and advisable to plug the found solution back into the original equation to verify it.

  • Fractional Exponents:

    • An expression like $(-Z+10)^{3/4} = 3$ can be solved by raising both sides to the reciprocal power, which is $4/3$. This cancels the original exponent, leaving $-Z+10 = 3^{4/3}$.

    • Alternatively, convert to radical form: the denominator of the exponent is the root (e.g., $4^{th}$ root), and the numerator is the power (e.g., $\sqrt[4]{(-Z+10)^3} = 3$).

    • Solving Approach:

      1. Isolate the term with the fractional exponent or radical.

      2. Raise both sides to a power to remove the root (e.g., raise to the $4^{th}$ power to remove a $4^{th}$ root), leaving the remaining exponent inside if applicable (e.g., leaving a $3^{rd}$ power).

      3. Raise both sides to the power corresponding to the remaining exponent (e.g., take the cube root to remove a $3^{rd}$ power).

    • Simplifying Radical Answers: If a cube root of $81$ is obtained, simplify it if possible (e.g., $3\sqrt[3]{3}$) but do not combine it with integers (e.g., $\sqrt[3]{81} - 10$ remains as is, rather than a decimal, unless specified).

Introduction to Module 4: Graphing

• Focus: Graphing linear equations.

• Linear Equations: Represent straight lines.

• Plotting Points:

  • Only two distinct points ($(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$) are needed to graph a straight line.

  • Choose convenient $x$ values (e.g., $0, -1, -5$) and calculate their corresponding $y$ values.

• ALEKS Graphing Tool:

  • The "line tool" can be used (pencil, eraser, line options).

  • Slope-Intercept Form ($y = mx+b$):

    • The $b$ value is the y-intercept, where the line crosses the y-axis (point $(0, b)$).

    • The $m$ value is the slope (rise over run).

    • Example: For $y = (1/3)x + 5$, the y-intercept is $(0, 5)$. From this point, rise $1$ unit (positive y-direction) and run $3$ units (positive x-direction) to find a second point.

  • "Plot a Point" Feature:

    • A very useful ALEKS tool that allows users to directly type in ($x, y$) coordinates.

    • This is especially helpful when exact placement on the grid is difficult, when coordinates are not whole numbers, or for more complex graphs in later units (parabolas, rational/radical graphs).

• Future Topics in Module 4 (to be covered on Thursday):

  • Distance formula and Midpoint formula.

  • Graphing vertical and horizontal lines.

  • Finding intercepts.

  • Real-world applications of linear equations.

  • Determining and calculating slopes (positive, negative, zero, undefined for vertical lines).

  • Parallel lines.