Question

The domain of \(\log (3-5 x)\) is (1) \(\left(\frac{3}{5}, \infty\right)\) (2) \(\left(0, \frac{3}{5}\right)\) (3) \(\left(-\infty, \frac{3}{5}\right)\) (4) \(\left(-\frac{3}{5}, 0\right)\)

Short answer

To summarize, the domain of the given logarithmic function \(\log (3-5x)\) is \(\left(-\infty, \frac{3}{5}\right)\), and the correct option among the given choices is: (3) \(\left(-\infty, \frac{3}{5}\right)\)

Step-by-step solution

Set the inequality

We have to find the values of x for which \(3-5x > 0\). Keep the inequality as it is and proceed to the next step.

Solve the inequality

Solve the inequality \(3-5x > 0\): Subtract 3 from both sides of the inequality to isolate the term with x: \(-5x > -3\) Now, divide both sides of the inequality by -5. Remember to reverse the inequality sign since we are dividing by a negative number: \(x < \frac{3}{5}\)

Identify the domain

Now, we have found that \(x < \frac{3}{5}\). It represents all the values of x that are less than \(\frac{3}{5}\). This means that the domain of the function is from negative infinity to \(\frac{3}{5}\). In interval notation, this is written as: \(\left(-\infty, \frac{3}{5}\right)\)

Match the domain to the given options

We found the domain to be \(\left(-\infty, \frac{3}{5}\right)\). Based on the given options, this corresponds to: (3) \(\left(-\infty, \frac{3}{5}\right)\)

Key concepts

Inequalities in Mathematics

Inequalities are mathematical expressions that compare two values, showing that one value is less than, greater than, less than or equal to, or greater than or equal to another. Inequalities are fundamental for determining ranges of possible solutions or conditions that numbers can satisfy. They are typically written using symbols such as < (less than), <= (less than or equal to), > (greater than), and >= (greater than or equal to).

Understanding inequalities helps in various fields of mathematics, including algebra, calculus, and even in the study of functions, where one must find the acceptable values that a variable can take on within a function, known as the domain. In the given exercise, we are examining an inequality within the context of a logarithmic function to establish its domain.

Inequalities play a crucial role in understanding how mathematical relationships aren't always equal and how sometimes we are tasked to find a range of solutions rather than a single answer. This patently applies when determining the intervals within which a function is defined or where a real-world situation is feasible.

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that make the inequality true. The process of solving inequalities is similar to solving equations but with a critical difference: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign reverses.

For instance, if we subtract a number from both sides of an inequality, the inequality sign remains the same. However, division by a negative number, such as in our exercise where we divided by -5, requires us to flip the inequality sign. This is a unique property of inequalities and is essential for accurate solutions.

Steps to Solve Inequalities:

• Isolate the variable on one side of the inequality.
• Simplify the inequality by performing the same mathematical operation on both sides.
• Keep in mind the special rule of reversing the inequality when multiplying or dividing by a negative number.
• Check the solution by plugging it back into the original inequality.

Interval Notation Mathematics

Interval notation is a method used in mathematics to describe a set of numbers lying between two endpoints. It's particularly useful when expressing the solution sets of inequalities and the domains or ranges of functions.

Interval notation consists of brackets and parentheses to indicate whether endpoints are included or excluded in the interval. For example, parentheses, \( ( \), are used for exclusion, and brackets, \[ [ \], for inclusion. When writing intervals, a parenthesis next to a number means the number is not part of the interval. In contrast, a bracket next to a number means the number is included.

For instance, in our textbook solution, \((-\infty, \frac{3}{5}\)) means all the numbers smaller than but not including \frac{3}{5}. Infinity symbols always come with parentheses since infinity itself is not a number and can't be reached or included.

Using interval notation allows concise representation of sets of real numbers and is a standard form when discussing continuous data, like domains of functions, making it an essential concept in higher-level mathematics.