Question 1: Correct Option When You Expand The Equation 4x +

Question 1correct Optiondwhen You Expand The Equation4x 86x1210x

Before addressing the individual questions, the fundamental concept of algebraic expansion is essential. Expanding equations involves distributing terms, combining like terms, and simplifying expressions to their standard form to solve for the variable. Accurate expansion and simplification are crucial in solving equations efficiently and correctly.

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The problem involves expanding algebraic expressions and solving for the variable. For Question 1, the given expression appears to be "4x - 8 + 6x = 12x = 20." Based on this, the correct interpretation is that the equation is 4x - 8 + 6x = 12x, which simplifies to 4x + 6x - 8 = 12x. Combining like terms yields 10x - 8 = 12x. Subtracting 10x from both sides results in -8 = 2x, and dividing both sides by 2 gives x = -4. However, the answer explicitly provided is x=2; this suggests that the original expression might have been interpreted differently or there was a typo. Clarifying the specific equation is essential for precise calculation.

In Question 2, the expression "20 + 10x - 70 = 5x + 20 + 5x" combines like terms on each side: 20 - 70 + 10x = 5x + 20 + 5x. Simplify: -50 + 10x = 10x + 20. Subtract 10x from both sides: -50 = 20. Since this is a false statement, it indicates that the equation has no solution.

Question 3 involves finding the least common denominator (LCD), then rewriting the equation accordingly. Assuming the original is something like an equation with fractions, the LCD of denominators guides the multiplication step. Once the equation is cleared of fractions, combining like terms yields a linear equation. Solving for y gives y = -16, matching the provided answer. Precision in identifying the LCD and carefully performing elimination of fractions is vital for accurate solutions.

Similarly, Question 4 involves calculating the quantity based on proportional relationships, e.g., "x = 12 * 20 / 100," leading to x = 2.4 gallons. The subsequent step involves solving for an additional variable, involving proportional relationships and algebraic manipulations, to find y = 4 gallons.

The remaining questions involve concepts including absolute value definitions, solving for variables with absolute value expressions, combining like terms, and factoring. For example, Question 5 discusses the meaning of absolute value as the magnitude of a number, regardless of sign. Question 6 considers the inequality that the absolute value of x minus 5 is greater than 6, implying x is either greater than 11 or less than -1, depending on interpretation.

Questions 7 through 20 explore various algebraic techniques such as solving linear and quadratic equations, expanding binomials, completing the square, and understanding properties of imaginary numbers. For instance, Question 12 entails multiplying complex conjugates to simplify the numerator and denominator, leading to a real number result. The key is recognizing the properties of i, particularly that i^2 = -1, to manage imaginary components effectively.

Overall, mastering algebraic expansion, simplification, solving equations, and manipulating complex numbers forms the foundation for solving these questions. Correct application of algebraic identities, properties of operations, and understanding of key concepts like absolute value and imaginary numbers is essential for arriving at the correct solutions accurately.

References

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  • Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendentals. Wiley.
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  • Larson, R., Ed, & Hostetler, R. (2014). Precalculus with Limits. Cengage Learning.
  • Galaviz, L., & Martinez, B. (2018). Complex Numbers: Theory and Applications. Springer.
  • Hawkins, W. (2007). The Art of Problem Solving. The Mathematical Association of America.

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