4.14 Unit Test: New Century Practice Quiz

Using the distributive property, multiply 2 by both x and 3 to get 2x + 6. This shows that each term inside the parentheses must be multiplied by 2.

Subtract 5 from both sides to isolate x, resulting in x = 5. This simple linear equation demonstrates basic algebraic manipulation.

The slope-intercept form of a linear equation is written as y = mx + b, where m represents the slope and b represents the y-intercept. This form is particularly useful for graphing linear equations.

Divide both the numerator and the denominator by their greatest common divisor, which is 4, to simplify 8/12 to 2/3. This results in the fraction in its simplest form.

Dividing 12 by 4 gives a quotient of 3. This is a fundamental arithmetic operation useful in many mathematical contexts.

Add 4 to both sides to obtain 2x = 14, then divide both sides by 2 to find x = 7. This problem reinforces the steps of solving a simple linear equation.

The distributive property means multiplying a by each term inside the parentheses: a(b + c) = ab + ac. This is a fundamental property used in algebra.

To factor x² + 5x + 6, look for two numbers that multiply to 6 and add up to 5, which are 2 and 3. This leads to the factorization (x+2)(x+3).

Multiply both sides of the equation by 2 to get 3y = 18, then divide by 3 to obtain y = 6. This problem requires manipulating an equation with a fractional coefficient.

The slope is calculated by the formula (y2 - y1) / (x2 - x1), which gives (11 - 3)/(5 - 2) = 8/3. This measures the rate of change between two points.

In the slope-intercept form y = mx + b, the term b represents the y-intercept. Here, b is -5, which means the line crosses the y-axis at -5.

The area of a triangle is calculated with the formula 1/2 � - base � - height. Substituting the values, 1/2 � - 8 � - 5 gives an area of 20 square units.

First, apply the distributive property to 4(x - 3) to obtain 4x - 12, then add 2x to get 6x - 12. Combining like terms is essential in simplifying algebraic expressions.

A standard deck has 52 cards, of which 26 are red, so the probability is 26/52, which simplifies to 1/2. This question assesses basic probability knowledge.

Cross-multiply to obtain 2 � - 9 = 3x, which simplifies to 18 = 3x. Dividing both sides by 3 gives x = 6.

First, distribute the multiplication on each side: 3x - 6 + 4 becomes 3x - 2, and 2(2x + 1) becomes 4x + 2. Rearranging the equation leads to x = -4.

Find a common denominator (12) to combine the fractions: (3x - 2x)/12 simplifies to x/12, and then multiply both sides by 12 to get x = 12. This problem tests fraction manipulation.

Let the smaller even integer be n, so the next consecutive even integer is n + 2. The equation n + (n + 2) = 46 simplifies to 2n + 2 = 46, yielding n = 22.

Let the width be w and the length be 3w. The area is given by 3w² = 48, so w² = 16 and the width w is 4. This problem applies both algebra and geometric reasoning.

Substitute x = 2 into the function: f(2) = 2(2²) - 3(2) + 1, which computes to 8 - 6 + 1 = 3. Evaluating the function at a specific input tests understanding of function notation.