Subtracting 3 from both sides gives 2x = 8, and dividing both sides by 2 yields x = 4. This process uses simple algebraic manipulation to isolate the variable.

Converting 1/2 to 2/4 gives 2/4 + 3/4, which equals 5/4. This addition of fractions demonstrates the importance of a common denominator.

The area of a rectangle is calculated by multiplying the length by the width, so 8 x 3 equals 24 square units. This basic geometric formula is fundamental in finding the area.

According to the order of operations, multiply 3 by 2 to get 6, then subtract it from 5 to obtain -1. This emphasizes the importance of performing multiplication before subtraction.

The slope is calculated as the change in y divided by the change in x: (6 - 2)/(3 - 1) equals 2. This question reinforces the concept of finding the rate of change between two points.

The quadratic factors as (x - 2)(x - 3) = 0, which yields the solutions x = 2 and x = 3. Factoring is a common method for solving basic quadratic equations.

Subtracting 2 from both sides gives 0.5x = 5, and dividing by 0.5 results in x = 10. This demonstrates how to solve a linear equation that includes decimal coefficients.

Distributing 2 over the parentheses gives 2x + 6, and subtracting x results in x + 6. Combining like terms is essential in simplifying algebraic expressions.

2³ equals 8 and 2² equals 4; their sum is 12. This problem reinforces the rules of exponentiation before performing addition.

Calculating 20% of 150 involves multiplying 150 by 0.2, which gives 30. This exercise tests understanding of percentages applied to real numbers.

Since the ratio is 3:5, for every 3 blue marbles there are 5 red marbles. With 15 blue marbles (which is 5 times 3), there must be 5 times 5 red marbles, totaling 25.

The interior angles of any triangle always add up to 180 degrees. This fundamental concept of geometry is essential in many mathematical problems.

By adding the equations, you obtain 2x = 14, so x = 7; then substituting back gives y = 3. This elimination method is efficient for solving simultaneous equations.

The expression x² - 9 is a difference of two squares and factors into (x - 3)(x + 3). Recognizing this algebraic identity is crucial in simplifying expressions.

Substituting x = 4 into the function gives f(4) = 3(4) - 5, which simplifies to 7. This question tests the evaluation of a linear function after input substitution.

Factoring the quadratic gives (x - 2)(x - 4) < 0, which is true when x is between 2 and 4. This means the expression is negative only in the interval 2 < x < 4.

Using the definition of logarithms, log₂(x) = 5 means that 2 to the power of 5 equals x. Therefore, x = 32, which is computed by 2❵.

Using the distance formula, the distance is √[(3 - (-1))² + (4 - 1)²] which simplifies to √(16 + 9) = √25 = 5. This calculation applies the Pythagorean theorem to coordinate points.

By using the substitution or elimination method, the system simplifies to give x = 27/11 and y = 26/11. This problem tests your ability to work with fractions and solve simultaneous equations.

The vertex of a quadratic function is found using the formula -b/(2a) for the x-coordinate, which in this case is 2. Substituting x = 2 back into the function produces f(2) = -1, so the vertex is (2, -1).