Subtracting 3 from both sides of the equation gives 2x = 4. Dividing both sides by 2 results in the solution x = 2.

According to the order of operations, multiplication is performed before addition. Multiplying 3 by 2 gives 6, and adding 5 results in 11.

The slope of a line in the slope-intercept form, y = mx + b, is given by m. Since parallel lines share the same slope, the answer is 3.

To add fractions with the same denominator, add the numerators. Therefore, 1/4 + 1/4 equals 2/4, which simplifies to 1/2.

Multiplying both sides of the equation by 2 isolates x. This yields x = 6 as the correct solution.

The equation x² - 9 = 0 is a difference of squares and can be factored into (x - 3)(x + 3) = 0. Setting each factor equal to zero gives the solutions x = 3 and x = -3.

The quadratic x² + 5x + 6 can be factored by finding two numbers that multiply to 6 and add to 5. These numbers are 2 and 3, which factor the expression as (x + 2)(x + 3).

Expanding the left-hand side gives 2x - 6 = x + 1. Subtracting x from both sides and then adding 6 leads to x = 7.

Completing the square for x² - 4x + 3 rewrites the quadratic as (x - 2)² - 1. This form makes it easy to identify the vertex at (2, -1).

For the square root function to be defined, the expression inside must be non-negative. Setting x - 2 ≥ 0 shows that x must be greater than or equal to 2.

By adding 5 to both sides of the inequality, we get 3x > 12. Dividing by 3 yields x > 4, which is the solution.

To find the inverse, swap x and y in the equation y = 2x + 3 and solve for y. This process results in y = (x - 3)/2, which is the inverse function.

Since f(x) squares the input, its inverse operation is taking the square root. Considering the domain restriction of x ≥ 0, the inverse function is f❻¹(x) = √x.

The logarithm log₃(27) asks for the exponent that produces 27 when 3 is raised to it. Since 3³ equals 27, the answer is 3.

Using the binomial theorem, the x² term in the expansion of (x + 2)³ is given by 3 times x² times 2. Multiplying 3 by 2 yields a coefficient of 6.

Solve the second equation x - y = 1 to obtain x = y + 1, then substitute into the first equation to get 2(y + 1) + 3y = 12. Simplifying leads to y = 2 and consequently x = 3.

Express the quadratic in vertex form as f(x) = a(x - 2)² - 3. Substituting the point (4, 5) into the equation gives 5 = a(2)² - 3, which simplifies to a = 2. Therefore, the correct form is 2(x - 2)² - 3.

Since the quadratic function opens downward due to the negative coefficient, its maximum value occurs at the vertex. The vertex is (1, 4), so the function values are all less than or equal to 4, making the range (-∞, 4].

The sum S of the first n terms of a geometric series is calculated using the formula S = a(1 - r❿)/(1 - r) for r ≠ 1. This formula accounts for the accumulated effect of the constant ratio over n terms.

First, evaluate g(2) which gives e². Applying the natural logarithm function f(x) = ln(x) to e², we obtain ln(e²) = 2. This demonstrates the inverse relationship between eˣ and ln(x).