Problem 43 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 3 x-15 \leq 30 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and express the solution in interval notation: \(3x - 15 \leq 30\) Answer: The solution of the given inequality is \(x \leq 15\), which can be expressed in interval notation as \((-\infty, 15]\).

Step 1: Isolate the variable on one side of the inequality

To isolate variable x on the left side of the inequality, we will add 15 to both sides of the inequality. $$ 3x - 15 + 15 \leq 30 + 15 $$ Simplifying the inequality we get: $$ 3x \leq 45 $$

Step 2: Solve the inequality

To solve the inequality, we will divide both sides of the inequality by 3. $$ \frac{3x}{3} \leq \frac{45}{3} $$ Simplifying the inequality we get: $$ x \leq 15 $$

Step 3: Write the solution in interval notation

The inequality \(x \leq 15\) in interval notation is: $$ (-\infty, 15] $$ This is the solution to the given inequality, representing that any value of x less than or equal to 15 is a valid solution.

Key Concepts

Algebra Basics

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It involves finding the unknown or putting real-life variables into equations. Inequalities are an essential part of algebra, where we find the range of values that satisfy certain conditions. In algebra, we often deal with expressions and equations. An expression is a combination of numbers, variables (like \( x \)), and operators (like \( + \), \( - \)). An equation, on the other hand, states that two expressions are equal.
This is typically represented using the \( = \) sign. In the context of inequalities, however, we use symbols like \( < \), \( > \), \( \leq \), and \( \geq \) to form an inequality statement. These inequalities require different techniques than solving standard equations, with one key method being the process of isolating the variable.

Variable Isolation

Variable isolation is a crucial step in solving equations and inequalities. The goal here is to make the variable of interest (for example \( x \) in our inequality) appear on one side of the inequality by itself. This is important for determining the solution set of the inequality. In the given exercise, the inequality began as \( 3x - 15 \leq 30 \).

• First, we added 15 to both sides of the inequality to counteract the \( -15 \) on the left. This led to \( 3x \leq 45 \).
• Next, we divided both sides by 3 in order to get \( x \) by itself, resulting in the inequality \( x \leq 15 \).

This two-step process of addition and division is a typical approach in variable isolation. It is essential to perform the same operation on both sides of the inequality to maintain the inequality's balance, similar to maintaining equation balance.

Understanding Interval Notation

Interval notation is a way of writing the set of all numbers between a pair of endpoints. It is commonly used in expressing the solution sets of inequalities. In interval notation, parentheses \(( )\) are used to indicate that an endpoint is not included, while square brackets \([ ]\) are used to indicate inclusion of an endpoint.
For instance, in the solution \( x \leq 15 \), the interval notation is \( (-\infty, 15] \). This means \( x \) can take any value less than or equal to 15, but not exceeding 15.
Here are some tips for using interval notation correctly:

• Use \([\) or \(]\) when a number is included in the set, often in the case of \( \leq \) or \( \geq \) signs.
• Use \((\) or \()\) for strict inequalities like \( < \) or \( > \).
• Always use \(-\infty\) or \(\infty\) with parentheses as they are concepts, not fixed values.

Providing Mathematical Solutions

Mathematical solutions are structured approaches to finding answers to given problems or equations. Solutions can be equations, inequalities, or any other mathematical expression that satisfies the conditions of a problem.
To solve the inequality \( 3x - 15 \leq 30 \), we followed a systematic approach:

• First, identify the mathematical problem and understand what is required. Here, it was to find the values of \( x \) satisfying the inequality.
• Proceed with variable isolation by performing operations that get \( x \) alone on one side of the inequality.
• Apply basic arithmetic operations like addition and division as needed.
• Finally, express the solution in a comprehensive form, such as interval notation, to clearly convey the range of values that satisfy the inequality.

It's important to keep solutions clear and concise to ensure they can be understood easily. Always verify solutions to ensure they satisfy the original conditions in the inequality problem.