Problem 25 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 53 x+12 \leq 13 $$

Step-by-Step Solution

Answer

The solution is \( x \leq \frac{1}{53} \) and in interval notation: \( (-\infty, \frac{1}{53}] \).

Step 1: Isolate the Variable Term

Begin by subtracting 12 from both sides of the inequality to start isolating the variable term:\[ 53x + 12 - 12 \leq 13 - 12 \]This simplifies to:\[ 53x \leq 1 \]

Step 2: Solve for x

Divide both sides of the inequality by 53, the coefficient of \(x\), to solve for \(x\):\[ \frac{53x}{53} \leq \frac{1}{53} \]This simplifies to:\[ x \leq \frac{1}{53} \]

Step 3: Verify and Translate Solution

Verify the solution by substituting values less than or equal to \(\frac{1}{53}\) into the original inequality to ensure it holds true. The inequality \( x \leq \frac{1}{53} \) is valid. Present the solution in interval notation:The solution set is \( \left(-\infty, \frac{1}{53}\right] \).

Step 4: Graph the Solution Set

To graph the solution, draw a number line. Shade the line to the left of \(\frac{1}{53}\) and include \(\frac{1}{53}\) by drawing a closed circle at this point, indicating that \(x\) can be \(\frac{1}{53}\) or any number less.

Key Concepts

Interval Notation

Interval notation is a way of representing the set of solutions for an inequality in a concise manner. It uses brackets to indicate which endpoints are included in the interval. In our specific problem, we found that the solutions for the inequality are all values of \( x \) such that \( x \leq \frac{1}{53} \).

• The interval starts from negative infinity (\(-\infty\)), as there's no lower bound for \( x \). This is represented by the parenthesis \((\), indicating that infinity is never actually included in the set.
• The interval ends at \(\frac{1}{53}\), which is included in the solution. Hence, a square bracket \([\) is used next to \(\frac{1}{53}\) to signify inclusion.

Combining these, we express the solution as \((-\infty, \frac{1}{53}]\). This tells us all the solutions are less than or equal to \(\frac{1}{53}\).

Number Line Graph

A number line graph is a simple, yet powerful, visual tool to represent solution sets of inequalities. This method allows you to easily see which numbers are included in the solution.
To graph the solution \( x \leq \frac{1}{53} \) on a number line:

• Draw a horizontal line and mark the point \(\frac{1}{53}\) on it.
• Use a closed circle on \(\frac{1}{53}\) to show that this number is included in the solution.
• Shade the line to the left of \(\frac{1}{53}\) to indicate all numbers less than \(\frac{1}{53}\) are part of the solution.

This graph indicates visually which values \( x \) can assume, making the solution more tangible and easy to understand.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions in order to solve for a variable. This process often requires following a sequence of steps: isolating variables, using inverse operations, and ensuring we maintain the balance of the equation or inequality.
For the inequality \( 53x + 12 \leq 13 \), here's a breakdown of the algebraic manipulation:

• First, subtract 12 from both sides of the inequality to remove the constant term next to the variable. This yields \( 53x \leq 1 \).
• Next, divide each side by 53 to isolate \( x \). This step provides the solution \( x \leq \frac{1}{53} \).

Throughout these steps, it's crucial to perform the same operation to both sides of the inequality to preserve its truth. When working with inequalities, always be cautious: for instance, if you multiply or divide by a negative number, you must reverse the inequality sign. However, in this exercise, we are only dealing with positive numbers, so the inequality direction remains unchanged. Solving this equation develops your understanding of how algebra can be applied to deduce all possible values a variable can take.