Solving 2x - 6 = 0, we add 6 to both sides to get 2x = 6 and then divide both sides by 2 to obtain x = 3. This straightforward process confirms that x = 3 is the correct answer.

Distributing the 3 gives 3x + 6, and subtracting 2x results in x + 6. This shows that the simplified expression is x + 6.

The equation y = 4x - 7 is in slope-intercept form, where the coefficient of x is the slope. Hence, the slope is 4.

A prime number is only divisible by 1 and itself. Since 7 meets this criterion, it is the prime number among the options.

Factorial of 5 means multiplying all positive integers from 1 to 5: 5 × 4 × 3 × 2 × 1 equals 120. This makes 120 the correct value for 5!.

Factoring the quadratic gives (x - 2)(x - 3) = 0, which means the solutions are x = 2 and x = 3. This method confirms that only these two numbers satisfy the equation.

Raising (2x²y) to the 3rd power gives 2³ multiplied by x^(2×3) multiplied by y³, resulting in 8x❶y³. This confirms the correct simplification.

To find the inverse, swap x and y in the equation y = 3x - 4 and solve for y, yielding y = (x + 4) / 3. This process confirms that the inverse function is (x + 4) / 3.

The discriminant is found using the formula b² - 4ac. Substituting a = 2, b = -4, and c = 1, we get 16 - 8 = 8, which is the correct value.

x² - 9 is a difference of squares and factors into (x - 3)(x + 3). This factorization directly confirms the correct answer.

Subtracting 3 from both sides gives 2x > 4 and then dividing by 2 results in x > 2. This confirms that the solution to the inequality is x > 2.

By rearranging the equation to slope-intercept form (y = mx + b), we find y = 4 - 2x. Setting x to 0 gives the y-intercept of 4.

Substitute x = 2 into the function to get f(2) = 2(4) - 3(2) + 1, which simplifies to 8 - 6 + 1 = 3. This calculation confirms the correct answer.

Subtracting 3y from both sides gives 2y - 7 = 9, and then adding 7 to both sides results in 2y = 16. Dividing by 2 yields y = 8, confirming the solution.

Finding a common denominator of 12, we convert 1/3 to 4/12 and 1/4 to 3/12; adding these gives 7/12. This shows the correct sum of the fractions is 7/12.

By solving x - y = 1, we express x as y + 1 and substitute into 2x + 3y = 12. This leads to a solution of y = 2 and x = 3, confirming the correct answer.

The expression inside the square root, x - 2, must be non-negative. Therefore, x must be at least 2, making the domain x ≥ 2.

The nth term of an arithmetic sequence is given by a₝ + (n - 1)d. Substituting a₝ = 5, d = 3, and n = 10, we get 5 + 9×3 = 32, which is the correct 10th term.

Squaring both sides of the equation removes the square root, yielding 2x + 3 = 25. Solving for x gives 2x = 22 and subsequently x = 11, which is the correct solution.

The definition of logarithms tells us that log₂(x) = 3 means x = 2³. Since 2³ equals 8, x is confirmed to be 8.