Algebra 1 Semester 2 Review Notes

Algebra 1 Semester 2 Review

System of Equations

• Problem: Mr. Frankel bought 7 tickets to a puppet show and spent $43. Child tickets cost $4 each, and adult tickets cost $9 each. Set up a system of equations to find the number of adult tickets ($a$) and child tickets ($c$) and solve for $a$.

• System of Equations:

  • Equation 1 (total tickets): $a + c = 7$

  • Equation 2 (total cost): $9a + 4c = 43$

• Solving for $a$: This will require solving the system of equations, likely using substitution or elimination.

Solving by Elimination

• Problem: Solve the following system using elimination:

  • $3x + 6y = 9$

  • $x - 6y = 11$

• Elimination Method: Adding the two equations will eliminate the $y$ variable, allowing us to solve for $x$.

Solving Systems of Equations

• Problem: Solve the following system of equations:

  • $3x - 4y = 21$

  • $4x + 2y = 6$

• Solution Method: Elimination or substitution can be used. For example, multiply the second equation by 2 to eliminate $y$.

Graphing Inequalities

• Problem: Graph the inequality: y < 7x - 9

• Graphing: Draw a dashed line at $y = 7x - 9$ (dashed because it's a strict inequality) and shade the region below the line (because $y$ is less than).

Solving Systems by Graphing

• Problem: Solve the system by graphing:

  • $x + y = 0$

  • $y = 4x + 45$

• Graphical Solution: Graph both lines. The solution is the point where the two lines intersect.

System of Inequalities

• Problem: Solve the system of inequalities graphically:

  • $y \leq 2x - 1$

  • x < -3

• Graphical Solution: Graph each inequality. The solution region is the area where the shaded regions of both inequalities overlap. The first inequality will have a solid boundary while the second will have a dashed one.

Simplifying Expressions with Exponents

• Problem: Simplify the expression. Leave the answer in exponent form. The actual expression is missing from the provided text.

• Example: $x^m \cdot x^n = x^{m+n}$, $(x^m)^n = x^{m \cdot n}$, $\frac{x^m}{x^n} = x^{m-n}$

Multiplication with Scientific Notation

• Problem: Multiply $(3.5 × 10^5)(8.2 × 10^2)$

• Multiplication: Multiply the coefficients and add the exponents: $(3.5 \cdot 8.2) × 10^{5+2} = 28.7 × 10^7$. This can be rewritten as $2.87 \cdot 10^8$ for proper scientific notation.

Simplifying Expressions with Variables and Exponents

• Problem: Simplify $(wc^7)(-8w^3c^5)$

• Simplification: Multiply the coefficients and add the exponents of like variables: $-8 \cdot w^{1+3} \cdot c^{1+5} = -8w^4c^6$

Power of a Power

• Problem: Simplify $(t^2v^{-4})^3$

• Simplification: Apply the power to each term inside the parentheses: $t^{2 \cdot 3}v^{-4 \cdot 3} = t^6v^{-12} = \frac{t^6}{v^{12}}$

Division with Exponents

• Problem: Simplify $\frac{(x^{+4})^9}{x^8}$

• Simplification: First, simplify the numerator: $x^{+4 \cdot 9}= x^{36}$. Then, divide: $\frac{x^{36}}{x^8} = x^{36-8} = x^{28}$

Arithmetic Sequences: Finding the nth Term

• Problem: Write a rule for the nth term of the arithmetic sequence: -10, -4, 2, 8, …

• Arithmetic Sequence: An arithmetic sequence has a constant difference between terms.

• Formula: $a<em>n = a</em>1 + (n-1)d$, where $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

• Solution: Here, $a<em>1 = -10$ and $d = -4 - (-10) = 6$. So, $a</em>n = -10 + (n-1)6 = -10 + 6n - 6 = 6n - 16$

Common Difference of an Arithmetic Sequence

• Problem: Find the common difference of the arithmetic sequence: 0, 0.4, 0.8, 1.2, …

• Common Difference: The constant value added to each term to get the next term.

• Solution: The common difference is $0.4 - 0 = 0.4$

Geometric Sequences: First Four Terms

• Problem: Give the first four terms of the geometric sequence for which $a_1 = -7$ and $r = -4$.

• Geometric Sequence: A sequence where each term is multiplied by a constant ratio to get the next term.

• Formula: $a<em>n = a</em>1 \cdot r^{n-1}$

• Solution:

  • $a_1 = -7$

  • $a_2 = -7 \cdot (-4) = 28$

  • $a_3 = 28 \cdot (-4) = -112$

  • $a_4 = -112 \cdot (-4) = 448$

  • The first four terms are -7, 28, -112, 448.

Exponential Decay Function

• Problem: Write an exponential function to model a population of 470 animals decreasing at an annual rate of 12%.

• Exponential Decay Formula: $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the rate of decay (as a decimal), and $t$ is time.

• Solution: $y = 470(1 - 0.12)^t = 470(0.88)^t$

Sum and Difference of Polynomials

• Problem: Find the sum or difference: $(6q^3 + 8q^2 + 3) + (8q^3 - 3q - 7)$

• Solution: Combine like terms:

  • $(6q^3 + 8q^3) + 8q^2 - 3q + (3 - 7) = 14q^3 + 8q^2 - 3q - 4$

Polynomials: Standard Form

• Problem: Write the polynomial so that the exponents decrease from left to right: $6x^3 - 6x + 4x^3 - 2$

• Solution: First combine like terms: $10x^3 - 6x - 2$, then write with decreasing exponents: $10x^3 - 6x - 2$

Classifying Polynomials

• Problem: Classify the expression $-9x^3 - 7$ and state its degree.

• Classification: Based on the description, it appears to be a binomial (two terms).

• Degree: The highest exponent of the variable is 3, so the degree is 3, making it a cubic binomial.

Multiplying Binomials

• Problem: Find the product: $(x + 4)(x + 7)$

• FOIL Method: Use the FOIL (First, Outer, Inner, Last) method to multiply the binomials.

• Solution: $x^2 + 7x + 4x + 28 = x^2 + 11x + 28$

Multiplying Polynomials

• Problem: Find the product: $(x + 5)(x^2 - 7x + 9)$

• Distribution: Distribute each term in the first polynomial to each term in the second polynomial.

• Solution: $x(x^2 - 7x + 9) + 5(x^2 - 7x + 9) = x^3 - 7x^2 + 9x + 5x^2 - 35x + 45 = x^3 - 2x^2 - 26x + 45$

Solving Polynomial Equations

• Problem: Solve the polynomial equation: $(x + 1)(2x + 6) = 0$

• Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero.

• Solution:

  • x + 1 = 0 => x = -1

  • 2x + 6 = 0 => 2x = -6 => x = -3

  • The solutions are $x = -1$ and $x = -3$

Factoring Polynomials

• Problem: Factor the polynomial: $x^2 + 7x + 12$

• Factoring: Find two numbers that multiply to 12 and add to 7.

• Solution: The numbers are 3 and 4. So, the factored form is $(x + 3)(x + 4)$

Simplifying Radical Expressions

• Problem: Simplify $\sqrt{56x^3}$

• Simplification: Factor out perfect squares from under the radical.

• Solution: $\sqrt{56x^3} = \sqrt{4 \cdot 14 \cdot x^2 \cdot x} = \sqrt{4x^2} \cdot \sqrt{14x} = 2x\sqrt{14x}$

Multiplying Radicals

• Problem: Simplify $\sqrt{10} \cdot \sqrt{4}$

• Simplification: $\sqrt{10} \cdot \sqrt{4} = \sqrt{10 \cdot 4} = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$

Combining Like Radicals

• Problem: Simplify $7\sqrt{6} + 8\sqrt{6} - 3\sqrt{6}$

• Simplification: Combine the coefficients of the like radicals.

• Solution: $(7 + 8 - 3)\sqrt{6} = 12\sqrt{6}$

Simplifying Radicals with Fractions

• Problem: Simplify $\sqrt{\frac27}$

• Simplification: $\sqrt{\frac27}=\frac{\sqrt{2}}{\sqrt7}=\frac{\sqrt{2}}{3}$

Probability: Taste Test

• Problem: In a taste test, 48 preferred the new soft drink, 112 preferred the old soft drink, and 40 could not tell any difference. What is the probability that a person chosen at random preferred the new soft drink?

• Probability: Probability = (Favorable Outcomes) / (Total Possible Outcomes)

• Solution:

  • Total participants: $48 + 112 + 40 = 200$

  • Probability of preferring the new soft drink: $\frac{48}{200} = \frac{12}{50} = \frac{6}{25}$

Probability: Rolling a Die

• Problem: What is the probability of rolling an even number or a 3 on a six-sided die?

• Probability:

  • Even numbers: 2, 4, 6 (3 outcomes)

  • Number 3: 3 (1 outcome)

  • Total favorable outcomes: 4

  • Total possible outcomes: 6

  • Probability = $\frac{4}{6} = \frac{2}{3}$

Experimental vs. Theoretical Probability

• Problem: A spinner with equal sections of blue, red, yellow, and green is spun 32 times. The results are in a table:

  • Red: 10

  • Green: 8

  • Blue: 7

  • Yellow: 7
    For which color is the experimental probability the same as the theoretical probability?

• Theoretical Probability: The theoretical probability for each color is $\frac{1}{4} = 0.25$

• Experimental Probabilities:

  • Red: $\frac{10}{32} = 0.3125$

  • Green: $\frac{8}{32} = 0.25$

  • Blue: $\frac{7}{32} = 0.21875$

  • Yellow: $\frac{7}{32} = 0.21875$

• Solution: The experimental probability of landing on Green is the same as its theoretical probability.

Probability Without Replacement

• Problem: A drawer contains 2 red socks, 7 white socks, and 9 blue socks. Without looking, you draw out a sock and then draw out a second sock without returning the first sock. What is the probability that the first sock and the second sock are both red?

• Probability Without Replacement: The outcome of the first event affects the probability of the second event.

• Solution:

  • Probability of first sock being red: $\frac{2}{18} = \frac{1}{9}$

  • After drawing one red sock, there is 1 red sock left and 17 total socks.

  • Probability of second sock being red (given the first was red): $\frac{1}{17}$

  • Probability of both socks being red: $\frac{1}{9} \cdot \frac{1}{17} = \frac{1}{153}$