Algebra 1 for US Curriculum

Algebra 1 is the backbone of Mathematics. It is usually taught in Grade-7 in the US. Noble Learners provides expert online tutoring for the USA from India. The basics of Algebra 1 are important for high school. Take a wise decision and become a part of our learning community. 
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Algebra 1 Syllabus

The chapters given below is the part of syllabus which we will discuss in course. In case of any variation in the syllabus our expert tutors from India will teach as per the need of the course from one student to another.

Chapter 1: The Foundation of Algebra

Chapter 2: Exploring the Realm of Real Numbers

Chapter 3: Navigating Linear Equations

Chapter 4: Visualizing Relations and Functions

Chapter 5: Unraveling Linear Equations

Chapter 6: Balancing Inequalities

Chapter 7: Mastering Systems of Linear Equations and Inequalities

Chapter 8: Unveiling the World of Polynomials

Chapter 9: The Art of Factoring

Chapter 10: Bridging Quadratic and Exponential Functions

Chapter 11: Unlocking Radical Expressions and Triangles

Chapter 12: Navigating Rational Expressions and Equations

Chapter 13: Exploring the World of Statistics

Chapter 14: Embracing the Odds in Probability

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Chapter 1: The Foundation of Algebra

  1. Question: Simplify \( \frac{3x^2 + 5x - 2}{x + 2} \).
    Answer: \( \frac{3x^2 + 5x - 2}{x + 2} = 3x - 1 \).
  2. Question: Solve the equation \( |2x - 3| = 7 \).
    Answer: \( x = \frac{10}{2} \) or \( x = -\frac{4}{2} \). So, \( x = 5 \) or \( x = -2 \).
  3. Question: Find the vertex of the quadratic function \( f(x) = x^2 - 4x + 3 \).
    Answer: The vertex is at \( \left(\frac{b}{2a}, f\left(\frac{b}{2a}\right)\right) = \left(\frac{4}{2}, f\left(\frac{4}{2}\right)\right) = (2, -1) \).
  4. Question: Expand and simplify \( (x - 2)^3 \).
    Answer: \( (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \).
  5. Question: Solve the inequality \( 2x^2 - 5x + 3 > 0 \).
    Answer: \( x \in (-\infty, \frac{1}{2}) \cup (3, \infty) \).

Chapter 2: Exploring the Realm of Real Numbers

  1. Question: Find all solutions to the equation \( \sqrt{3x + 4} - 2 = 0 \).
    Answer: \( x = \frac{4}{3} \).
  2. Question: Determine if \( \frac{5}{6} \) is a rational or irrational number.
    Answer: \( \frac{5}{6} \) is a rational number.
  3. Question: Simplify \( \sqrt{72} \).
    Answer: \( \sqrt{72} = 6\sqrt{2} \).
  4. Question: Solve the equation \( x^3 + 8 = 0 \).
    Answer: \( x = -2 \).
  5. Question: Find the value of \( x \) in \( |2x - 1| = 5 \).
    Answer: \( x = 3 \) or \( x = -2 \).

Chapter 3: Navigating Linear Equations

  1. Question: Solve the system of equations: \( \begin{cases} 2x - y = 5 \\ x + 3y = 2 \end{cases} \).
    Answer: \( x = -\frac{11}{7} \) and \( y = -\frac{9}{7} \).
  2. Question: Determine the slope and y-intercept of the line \( 3x + 4y = 12 \).
    Answer: Slope = \( -\frac{3}{4} \) and y-intercept = 3.
  3. Question: Find the equation of the line passing through the points (2, 3) and (4, 5).
    Answer: \( y = x + 1 \).
  4. Question: Solve the inequality \( 2x - 3 \leq 4x + 5 \).
    Answer: \( x \geq -4 \).
  5. Question: Determine if the point (2, 3) lies on the line \( y = -2x + 7 \).
    Answer: Yes, it lies on the line.

Chapter 4: Visualizing Relations and Functions

  1. Question: Determine if the relation represented by the points (1, 4), (2, 6), (3, 8), (4, 10) is a function.
    Answer: Yes, it is a function.
  2. Question: Find the domain and range of the function \( f(x) = \frac{1}{x} \).
    Answer: Domain is \( x \neq 0 \) and range is \( y \neq 0 \).
  3. Question: Sketch the graph of the function \( y = \sqrt{x + 2} \).
    Answer: The graph is a square root function shifted 2 units to the left.
  4. Question: Find the inverse of the function \( f(x) = 2x + 3 \).
    Answer: \( f^{-1}(x) = \frac{x - 3}{2} \).
  5. Question: Determine if the function \( f(x) = x^3 + 2x \) is even, odd, or neither.
    Answer: Odd.

Chapter 5: Unraveling Linear Equations

  1. Question: Solve the equation \( 3(x + 4) - 2(2x - 3) = 10 \).
    Answer: \( x = \frac{11}{7} \).
  2. Question: Simplify \( \frac{2x^2 + 3x - 1}{x^2 - 4} \).
    Answer: \( \frac{2x^2 + 3x - 1}{x^2 - 4} = \frac{(2x - 1)(x + 1)}{(x - 2)(x + 2)} \).
  3. Question: Find the slope of the line passing through the points (2, -3) and (-4, 5).
    Answer: Slope = \( -\frac{4}{3} \).
  4. Question: Determine if the lines \( 3x - 2y = 5 \) and \( 6x - 4y = 10 \) are parallel, perpendicular, or neither.
    Answer: Parallel.
  5. Question: Solve the inequality \( -2x + 3 \geq 7 - 4x \).
    Answer: \( x \leq -2 \).

Chapter 6: Balancing Inequalities

  1. Question: Solve the inequality \( 2x + 5 \geq 3x - 2 \).
    Answer: \( x \leq 7 \).
  2. Question: Determine the solution set of \( |2x - 3| < 5 \).
    Answer: \( -1 < x < 4 \).
  3. Question: Simplify \( 3(2x - 4) > 2(3x - 5) \).
    Answer: \( x > \frac{2}{3} \).
  4. Question: Solve the compound inequality \( -4 \leq 2x + 3 < 8 \).
    Answer: \( -\frac{7}{2} \leq x < \frac{5}{2} \).
  5. Question: Determine the values of \( x \) that satisfy \( x^2 - 9 > 0 \).
    Answer: \( x < -3 \) or \( x > 3 \).

Chapter 7: Mastering Systems of Linear Equations and Inequalities

  1. Question: Solve the system of equations: \( \begin{cases} 2x - 3y = 7 \\ x + y = 5 \end{cases} \).
    Answer: \( x = 4 \) and \( y = 1 \).
  2. Question: Determine if the system of inequalities \( \begin{cases} 3x + 2y \leq 6 \\ 2x - y \geq 1 \end{cases} \) has a solution.
    Answer: Yes, it has a solution.
  3. Question: Find the solution set of \( \begin{cases} x + y < 4 \\ 2x - 3y > 6 \end{cases} \).
    Answer: \( (-\infty, 4) \) for \( x \) and \( (-\infty, \frac{2}{3}x - 2) \) for \( y \).
  4. Question: Solve the system of equations and inequalities: \( \begin{cases} x + y = 3 \\ x - y \geq 1 \end{cases} \).
    Answer: \( x = 2 \) and \( y = 1 \).
  5. Question: Determine if the lines represented by the equations \( 2x + 3y = 6 \) and \( 4x + 6y = 12 \) are parallel, perpendicular, or neither.
    Answer: They are the same line, so they are parallel.

Chapter 8: Unveiling the World of Polynomials

  1. Question: Factor the polynomial \( x^2 - 4x + 4 \).
    Answer: \( (x - 2)^2 \).
  2. Question: Find the roots of the quadratic equation \( x^2 - 5x + 6 = 0 \).
    Answer: \( x = 2 \) or \( x = 3 \).
  3. Question: Simplify the expression \( (x^2 - 3x + 2)(x^2 + 3x + 2) \).
    Answer: \( x^4 - x^2 - 4 \).
  4. Question: Factor completely: \( 2x^3 - 8x^2 + 8x \).
    Answer: \( 2x(x - 2)^2 \).
  5. Question: Determine if \( x - 1 \) is a factor of \( x^3 - 2x^2 + 3x - 4 \).
    Answer: No, \( x - 1 \) is not a factor.

Chapter 9: The Art of Factoring

  1. Question: Factor the expression \( 4x^2 - 9 \).
    Answer: \( (2x + 3)(2x - 3) \).
  2. Question: Factor the expression \( x^4 - 16 \).
    Answer: \( (x^2 - 4)(x^2 + 4) \).
  3. Question: Find all the factors of the polynomial \( x^3 - 8x^2 + 15x - 6 \).
    Answer: \( (x - 1)(x - 2)(x - 3) \).
  4. Question: Factor completely: \( 3x^3 + 6x^2 - 15x \).
    Answer: \( 3x(x - 1)(x + 5) \).
  5. Question: Determine if \( x - 2 \) is a factor of \( 2x^3 - 7x^2 + 4x - 8 \).
    Answer: Yes, \( x - 2 \) is a factor.

Chapter 10: Bridging Quadratic and Exponential Functions

  1. Question: Solve the quadratic equation \( x^2 - 5x + 6 = 0 \).
    Answer: \( x = 2 \) or \( x = 3 \).
  2. Question: Find the value of \( x \) in \( 2^x = 8 \).
    Answer: \( x = 3 \).
  3. Question: Sketch the graph of the function \( y = 2^x \).
    Answer: Exponential growth function.
  4. Question: Find the vertex of the quadratic function \( f(x) = x^2 - 4x + 3 \).
    Answer: Vertex is at \( (2, -1) \).
  5. Question: Determine if the function \( f(x) = x^2 + 2x + 1 \) is quadratic or linear.
    Answer: Quadratic.

Chapter 11: Unlocking Radical Expressions and Triangles

  1. Question: Simplify \( \sqrt{50} \).
    Answer: \( 5\sqrt{2} \).
  2. Question: Find the value of \( x \) in the equation \( \sqrt{x + 3} = 5 \).
    Answer: \( x = 22 \).
  3. Question: Solve the right triangle with a hypotenuse of 10 and one leg of length 6.
    Answer: The other leg is \( 8 \) and the angles are approximately \( 36.87^\circ \) and \( 53.13^\circ \).
  4. Question: Simplify \( \sqrt{18} - \sqrt{8} \).
    Answer: \( 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \).
  5. Question: Find the length of the diagonal of a square with side length \( 5 \).
    Answer: \( 5\sqrt{2} \).

Chapter 12: Navigating Rational Expressions and Equations

  1. Question: Simplify \( \frac{3x^2 + 5x - 2}{2x^2 - x - 3} \).
    Answer: \( \frac{(3x - 1)(x + 2)}{(2x - 3)(x + 1)} \).
  2. Question: Solve the equation \( \frac{x - 1}{x + 2} = \frac{3}{2} \).
    Answer: \( x = \frac{5}{2} \).
  3. Question: Find the domain of the function \( f(x) = \frac{2x}{x^2 - 4} \).
    Answer: \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).
  4. Question: Simplify \( \frac{6}{x^2 - 4} + \frac{3}{x - 2} \).
    Answer: \( \frac{9}{x^2 - 4} \).
  5. Question: Solve the rational inequality \( \frac{x - 2}{x + 1} \geq 0 \).
    Answer: \( x \in (-\infty, -1) \cup [2, \infty) \).

Chapter 13: Exploring the World of Statistics

  1. Question: Calculate the mean of the following data set: \( 10, 15, 20, 25, 30 \).
    Answer: Mean = \( \frac{10 + 15 + 20 + 25 + 30}{5} = 20 \).
  2. Question: Find the median of the data set: \( 5, 7, 12, 15, 20, 25, 30 \).
    Answer: Median = \( 15 \).
  3. Question: Calculate the standard deviation of the data set: \( 3, 5, 8, 12, 15 \).
    Answer: Standard deviation ≈ \( 4.18 \).
  4. Question: Find the mode of the data set: \( 7, 7, 10, 12, 15, 15, 15, 20 \).
    Answer: Mode = \( 15 \).
  5. Question: Calculate the range of the data set: \( 10, 15, 20, 25, 30 \).
    Answer: Range = \( 20 \).

Chapter 14: Embracing the Odds in Probability

  1. Question: Calculate the probability of rolling a 5 on a fair six-sided die.
    Answer: Probability = \( \frac{1}{6} \).
  2. Question: If you have a bag containing 3 red, 2 blue, and 5 green marbles, what is the probability of drawing a red marble?
    Answer: Probability = \( \frac{3}{10} \).
  3. Question: A deck of cards contains 52 cards. What is the probability of drawing a heart?
    Answer: Probability = \( \frac{1}{4} \).
  4. Question: If you flip a fair coin twice, what is the probability of getting exactly one head?
    Answer: Probability = \( \frac{1}{2} \).
  5. Question: Two dice are rolled. What is the probability of getting a sum of 7?
    Answer: Probability = \( \frac{1}{6} \).
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