뒤로Precalculus Test I: Step-by-Step Guidance and Solutions
스터디 가이드 - 스마트 노트
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Q1. Solve: $23x - 5 = 12x - 3$. Write your answer in solution set notation.
Background
Topic: Solving Linear Equations
This question tests your ability to solve a basic linear equation for the variable $x$ and express the solution in set notation.
Key Terms and Formulas:
Linear Equation: An equation of the form $ax + b = cx + d$.
Solution Set Notation: The set of all solutions, usually written as $\{x\}$.
Step-by-Step Guidance
Start by moving all terms involving $x$ to one side of the equation and constants to the other side.
Combine like terms to simplify the equation.
Isolate $x$ by dividing both sides by the coefficient of $x$.
Express your answer in solution set notation: $\{x\}$.
Try solving on your own before revealing the answer!
Final Answer: $\left\{\dfrac{2}{11}\right\}$
After simplifying and solving for $x$, you get $x = \dfrac{2}{11}$. In solution set notation, this is $\left\{\dfrac{2}{11}\right\}$.
Q2. Solve the following formula for $n$: $E = z \cdot s n$.
Background
Topic: Literal Equations (Solving for a Variable)
This question asks you to rearrange a formula to solve for a specific variable, $n$.
Key Terms and Formulas:
Literal Equation: An equation involving two or more variables.
Goal: Isolate $n$ on one side of the equation.
Step-by-Step Guidance
Identify the variable you need to solve for ($n$).
Notice that $z$ and $s$ are multiplied by $n$.
Divide both sides of the equation by $z \cdot s$ to isolate $n$.
Write the resulting expression for $n$ in terms of $E$, $z$, and $s$.
Try solving on your own before revealing the answer!
Final Answer: $n = \dfrac{E}{z \cdot s}$
Dividing both sides by $z \cdot s$ isolates $n$.
Q3. Solve by factoring and using the Zero-Product Property: $12x^2 - 41x + 24 = 0$. Write your solution in solution set notation.
Background
Topic: Solving Quadratic Equations by Factoring
This question tests your ability to factor a quadratic equation and use the Zero-Product Property to find solutions.
Key Terms and Formulas:
Quadratic Equation: $ax^2 + bx + c = 0$
Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.
Factoring: Expressing the quadratic as a product of two binomials.
Step-by-Step Guidance
Write the quadratic equation in standard form: $12x^2 - 41x + 24 = 0$.
Look for two numbers that multiply to $12 \times 24$ and add to $-41$.
Use factoring by grouping or trial and error to factor the quadratic into two binomials.
Set each factor equal to zero and solve for $x$.
Express your solutions in solution set notation: $\{x_1, x_2\}$.
Try solving on your own before revealing the answer!
Final Answer: $\left\{\dfrac{24}{12}, \dfrac{17}{12}\right\}$ or $\left\{2, \dfrac{17}{12}\right\}$
Factoring gives $(12x - 9)(x - 2) = 0$, so $x = 2$ and $x = \dfrac{9}{12} = \dfrac{3}{4}$. However, check your factoring for accuracy. The correct factors are $(12x - 9)(x - 2) = 0$ leading to $x = 2$ and $x = \dfrac{9}{12} = \dfrac{3}{4}$.
Q4. Find the real solutions, if any, to $2x^2 - 6x + 1 = 0$ using the Quadratic Formula. Write your solution in solution set notation.
Background
Topic: Quadratic Formula
This question tests your ability to use the quadratic formula to solve a quadratic equation.
Key Terms and Formulas:
Quadratic Formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $b^2 - 4ac$ (determines the nature of the roots)
Step-by-Step Guidance
Identify $a = 2$, $b = -6$, and $c = 1$ from the equation.
Plug these values into the quadratic formula.
Calculate the discriminant $(-6)^2 - 4 \cdot 2 \cdot 1$.
Write the solutions in the form $x = \dfrac{-b \pm \sqrt{\text{discriminant}}}{2a}$.
Express your answers in solution set notation.
Try solving on your own before revealing the answer!
Final Answer: $\left\{3 + \dfrac{\sqrt{28}}{4}, 3 - \dfrac{\sqrt{28}}{4}\right\}$ or $\left\{3 + \dfrac{\sqrt{28}}{4}, 3 - \dfrac{\sqrt{28}}{4}\right\}$
Plugging into the quadratic formula gives two real solutions. Simplify $\sqrt{28}$ as needed.
Q5. Use the discriminant to determine the nature of the solutions for each quadratic equation (do not solve):
(a) $4x^2 + 12x + 9 = 0$
(b) $3x^2 - 2x + 10 = 0$
Background
Topic: Discriminant and Nature of Quadratic Roots
This question tests your ability to use the discriminant $b^2 - 4ac$ to determine if a quadratic has two real solutions, one real solution, or no real solutions.
Key Terms and Formulas:
Discriminant: $D = b^2 - 4ac$
If $D > 0$: Two unequal real solutions
If $D = 0$: One repeated real solution
If $D < 0$: No real solutions
Step-by-Step Guidance
For each equation, identify $a$, $b$, and $c$.
Calculate the discriminant $D = b^2 - 4ac$ for each.
Interpret the value of $D$ to determine the nature of the solutions.
State whether there are two real solutions, one real solution, or no real solutions for each equation.
Try solving on your own before revealing the answer!
Final Answer:
(a) $D = 0$ → One repeated real solution.
(b) $D < 0$ → No real solutions.
For (a), the discriminant is zero, so there is one real solution. For (b), the discriminant is negative, so there are no real solutions.
Q6. Solve: $2x + 5 = x - 5$. Write your solution in solution set notation.
Background
Topic: Solving Linear Equations
This question tests your ability to solve a simple linear equation and express the answer in set notation.
Key Terms and Formulas:
Linear Equation: An equation of the form $ax + b = cx + d$.
Solution Set Notation: $\{x\}$.
Step-by-Step Guidance
Move all terms involving $x$ to one side and constants to the other.
Combine like terms to isolate $x$.
Write your answer in solution set notation.
Try solving on your own before revealing the answer!
Final Answer: $\{ -10 \}$
After isolating $x$, you find $x = -10$.
Q7. Solve by factoring: $3x^4 - 48x^2 = 0$. Write your solution in solution set notation.
Background
Topic: Factoring and Solving Polynomial Equations
This question tests your ability to factor a quartic equation and use the Zero-Product Property to find all real solutions.
Key Terms and Formulas:
Factoring: Expressing the equation as a product of factors.
Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.
Step-by-Step Guidance
Factor out the greatest common factor from the equation.
Recognize the remaining expression as a quadratic in $x^2$ and factor further if possible.
Set each factor equal to zero and solve for $x$.
List all real solutions in solution set notation.
Try solving on your own before revealing the answer!
Final Answer: $\{0, 4, -4\}$
Factoring gives $3x^2(x^2 - 16) = 0$, so $x = 0, 4, -4$.
Q8. Solve: $-4x + 2 \geq 3x + 20$. Write your solution in interval notation.
Background
Topic: Solving Linear Inequalities
This question tests your ability to solve a linear inequality and express the solution in interval notation.
Key Terms and Formulas:
Linear Inequality: An inequality involving a linear expression.
Interval Notation: Expressing the set of solutions as an interval on the real number line.
Step-by-Step Guidance
Move all terms involving $x$ to one side and constants to the other.
Combine like terms to isolate $x$.
When dividing or multiplying both sides by a negative number, remember to reverse the inequality sign.
Express the solution in interval notation.
Try solving on your own before revealing the answer!
Final Answer: $(-\infty, -\dfrac{18}{7}]$
After solving, the solution is all $x$ less than or equal to $-\dfrac{18}{7}$.
Q9. Solve: $2x - 3 = 11$. Write your solution in set notation.
Background
Topic: Solving Linear Equations
This question tests your ability to solve a simple linear equation and express the answer in set notation.
Key Terms and Formulas:
Linear Equation: $ax + b = c$
Set Notation: $\{x\}$
Step-by-Step Guidance
Add $3$ to both sides to isolate the $2x$ term.
Divide both sides by $2$ to solve for $x$.
Write your answer in set notation.
Try solving on your own before revealing the answer!
Final Answer: $\{7\}$
Solving gives $x = 7$.
Q10. Solve: $3x + 5 < 17$. Write your solution in interval notation.
Background
Topic: Solving Linear Inequalities
This question tests your ability to solve a linear inequality and express the solution in interval notation.
Key Terms and Formulas:
Linear Inequality: $ax + b < c$
Interval Notation: $(a, b)$
Step-by-Step Guidance
Subtract $5$ from both sides to isolate the $3x$ term.
Divide both sides by $3$ to solve for $x$.
Express the solution in interval notation.
Try solving on your own before revealing the answer!
Final Answer: $(-\infty, 4)$
After solving, the solution is all $x$ less than $4$.
