Problem 1 - Elementary Algebra

Question

Graph all solutions on a number line and provide the corresponding interval notation. $$ x \leq 10 $$

Step-by-Step Solution

Answer

The interval notation is \((- abla, 10]\).

Step 1: Understand the Inequality

The inequality given is \(x \leq 10\), which means \(x\) can be any number less than or equal to 10.

Step 2: Identify the Type of Solution

For the inequality \(x \leq 10\), the solutions include all numbers less than or equal to 10. This will be a continuous set of numbers on the number line.

Step 3: Draw the Number Line

On a number line, place a closed circle at 10 to indicate that 10 is included in the solutions. Shade the line to the left of 10, covering all numbers less than 10.

Step 4: Write the Interval Notation

To express the solutions in interval notation, start with negative infinity \((\infty)\) and end with 10. Use a bracket to include 10 since \(x\) can be equal to 10. Thus, the interval is \((-abla, 10]\).

Key Concepts

Number Line

A number line is a visual representation of numbers in order. It helps us understand and solve inequalities by showing how numbers relate to each other spatially.
On a number line, numbers increase as you move to the right and decrease to the left.

• To represent the inequality \(x \leq 10\) on a number line, we show that \(x\) includes all numbers less than or equal to 10.
• We use a closed circle on the number "10" to indicate that 10 is a part of the solution. This shows that the value is included in the set.
• Shade or draw a line to the left of the closed circle to display all the numbers that are solutions to the inequality \(x \leq 10\).

Using a number line is helpful when graphing solutions, as it provides a clear picture of which numbers satisfy the inequality.

Interval Notation

Interval notation is a shorthand way to express a range of numbers, and it's especially useful for describing the solutions to inequalities. It allows us to communicate the set of numbers that are the solution without drawing a full graph.
For the inequality \(x \leq 10\), we use interval notation as follows:

• Since there is no starting number and \(x\) can be any value up to 10, we begin at negative infinity \((-\infty)\).
• The number 10 is included, so we use a bracket \(\left[\right]\) to show inclusion: \( 10 ]\).

Together, the interval notation for \(x \leq 10\) is \((-\infty, 10]\). This notation is concise and convenient for describing an endless set of numbers.

Graphing Inequalities

Graphing inequalities involves visually representing all possible solutions of an inequality on a number line or coordinate plane, which makes it easier to understand and interpret.
To graph an inequality like \(x \leq 10\), follow these steps:

• Start by drawing a basic number line, with numbers proportionally placed.
• Locate the number 10 on the line and place a closed circle there. The closed circle shows that 10 is included in our solution set.
• Draw a shaded region or line extending to the left of 10, symbolizing all numbers that are less than 10.

This graphical representation clarifies all potential values of \(x\) that satisfy the inequality and allows for easy identification of solution sets.