Basic Arithmetic and Number Systems

The mean is the average of a dataset calculated by summing all values and dividing by the number of values. The median is the middle value when the data is arranged in order. The mode is the most frequently occurring value in a dataset. For example, consider the dataset: 2, 4, 5, 5, 6, 7, 9. The mean is 5, the median is 5, and the mode is 5.

The domain of $f(x) = \frac{1}{x-2}$ is all real numbers except for $x=2$. This is because when $x=2$, the denominator becomes zero, which is undefined. To determine the domain, we need to identify any values of $x$ that would make the denominator zero.

Trigonometry is widely used in navigation to determine distances and directions based on angles and side lengths. It is also essential in surveying to accurately map land features and calculate distances and areas.

<p>The formula $\frac{1}{2}bh$ for the area of a triangle is derived from the fact that a triangle is exactly half of a parallelogram with the same base and height. The area of a parallelogram is $bh$, and since a triangle is half of it, the area of a triangle is $\frac{1}{2}bh$.</p> Signup and view all the answers

<p>A function is a relationship between two sets of variables, where each input (x-value) has a unique output (y-value). Key characteristics include the domain, range, intercepts, and asymptotes. The domain is the set of all possible inputs, the range is the set of all possible outputs, intercepts are points where the function crosses the axes, and asymptotes are lines that the function approaches as the input goes to infinity.</p> Signup and view all the answers

<p>One scenario where subtraction is used in daily life is when calculating the remaining balance after making a purchase. For example, if you have $50 and buy groceries for $35, you would use subtraction to find the amount of money left ($50 - $35 = $15). Subtraction is appropriate because it represents the removal of a value from an initial whole.</p> Signup and view all the answers

<p>Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero. For example, 3/4 is a rational number. Irrational numbers cannot be expressed as a fraction and their decimal representations are non-repeating and non-terminating. An example of an irrational number is pi (π).</p> Signup and view all the answers

<p>Factoring in algebra is the process of breaking down an expression into simpler expressions, typically by finding common factors. This can be helpful in solving equations by simplifying them or finding solutions. For example, to solve the equation $x^2 - 4 = 0$, we can factor it into (x + 2)(x - 2) = 0, which leads to the solutions x = -2 and x = 2.</p> Signup and view all the answers

<p>Congruence in geometry means that two shapes are identical in both size and shape. They have the same corresponding side lengths and angles. An example of two congruent shapes are two equilateral triangles with side lengths of 5 cm each. They have the same shape and their sides and angles are equal.</p> Signup and view all the answers

<p>Translation involves moving a shape to a new location without changing its size or orientation. It is like sliding the shape along a straight path. Reflection, on the other hand, produces a mirror image of a shape. It flips the shape across a line, maintaining its size but reversing its orientation.</p> Signup and view all the answers

<p>Units of measurement provide a common standard for quantifying physical properties like length, mass, volume, and time. Standard units are essential because they ensure consistency and prevent confusion in communication. For instance, using meters instead of feet for measuring distance allows everyone to understand the same measurement.</p> Signup and view all the answers

<p>A polynomial in algebra is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The exponents of the variables must be nonnegative integers. An example of a polynomial expression is 3x^2 + 2x - 5.</p> Signup and view all the answers

<p>An inequality in algebra represents a relationship where two expressions are not equal. Instead of using an equal sign (=), it uses symbols like &lt; (less than), &gt; (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Understanding inequalities is crucial for solving problems that involve comparisons, ranges of values, or constraints.</p> Signup and view all the answers