Questions

Solve for \( x \) in the linear equation \( 3x + 7 = 22 \)

Keywords: solve for x, linear equation, 3x + 7 = 22, algebra, solution steps
This problem shows how to solve the linear equation \( 3x + 7 = 22 \) for \( x \). The solution involves using the properties of equality to isolate \( x \), resulting in \( x = 5 \). It covers concepts of linear equations in one variable, properties of equality, and algebraic manipulation.

Simplify Polynomial Expression (4x³ - 2x² + 5x - 1) - (x³ + 3x² - 2x + 4)

Keywords: polynomial simplification, combining like terms, polynomial subtraction, distributive property, middle school math
Step-by-step solution to simplify the polynomial expression \((4x^3 - 2x^2 + 5x - 1) - (x^3 + 3x^2 - 2x + 4)\) using the distributive property (subtraction rule) and combining like terms, resulting in \(3x^3 - 5x^2 + 7x - 5\).

Balancing CH₄ + Cl₂ → CCl₄ + HCl Chemical Equation

Keywords: balancing chemical equation, CH₄ Cl₂ CCl₄ HCl, stoichiometric coefficients, law of conservation of mass, grade 9 chemistry
This problem requires balancing the chemical equation CH₄ + Cl₂ → CCl₄ + HCl. The solution uses element conservation (C, H, Cl) to determine coefficients: 1 CH₄, 4 Cl₂, 1 CCl₄, and 4 HCl, following the Law of Conservation of Mass.

Find the slope of the line through (-2,1) and (3,1)

Keywords: slope of line, slope formula, horizontal line slope, middle school math, two points slope
This problem is about finding the slope of a line passing through the points (-2, 1) and (3, 1) in coordinate geometry. It is solved using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), and the resulting slope is 0.

Simplify \( 5\sqrt{96} - 7\sqrt{54} \)

Keywords: simplify radicals, combining like radicals, square roots, high school math
Step-by-step solution to simplifying \( 5\sqrt{96} - 7\sqrt{54} \) by factoring radicands and combining like radicals.

Lake Water Level Decreasing 3% Weekly: Find Level After 8 Weeks

Keywords: exponential decay, percent decrease, lake water level, high school math
Step-by-step solution to find lake water level after 8 weeks of 3% weekly decrease using the exponential decay formula.

Expand \( \log_2 \frac{x^5}{8} \) Using Logarithmic Properties

Keywords: expand logarithm, logarithmic properties, quotient rule, power rule, high school math
Step-by-step expansion of \( \log_2 \frac{x^5}{8} \) using the quotient and power rules of logarithms.

Solve \( -4 \ln(6x) = 2 \) Using Natural Logarithm Inverse Property

Keywords: natural logarithm, logarithmic equation, inverse property, high school math
Step-by-step solution to solve \( -4 \ln(6x) = 2 \) by isolating the logarithm and using the inverse property of \( e \) and \( \ln \).

Discontinuities of Rational Function \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \)

Keywords: rational function, discontinuities, vertical asymptote, hole, factoring polynomials, high school math
Step-by-step analysis of discontinuities in \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \) by factoring numerator and denominator.

Subtract and Simplify \( \frac{5x}{3x^2} - \frac{7}{6x} \)

Keywords: subtract rational expressions, least common denominator, simplify fractions, high school math
Step-by-step solution to subtracting \( \frac{5x}{3x^2} - \frac{7}{6x} \) by finding LCD and simplifying the result.

Solve \( \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} \) and Check for Extraneous Solution

Keywords: rational equation, extraneous solution, factoring trinomials, LCD, high school math
Step-by-step solution to solving the rational equation \( \frac{5}{a - 4} + \frac{2}{a + 1} = \frac{-31}{a^2 - 3a - 4} \) and checking for extraneous solutions.

Express tan B as a Simplest Fraction in a Right Triangle

Keywords: tan B, right triangle, tangent ratio, simplest fraction, radical, adjacent side, opposite side
This problem involves finding \( \tan B \) in a right triangle (right-angled at \( C \)) using the tangent ratio (opposite/adjacent). The solution identifies the sides opposite and adjacent to \( \angle B \), applies the tangent formula, and simplifies the fraction.