Problem 91 - Algebra 1

Question

Solve the inequality and graph the solution. 2 \leq x<5

Step-by-Step Solution

Answer

The solution for the inequality is x belongs to the interval [2, 5)

Step 1: Understand the inequality

The inequality '2 ≤ x < 5' says that x is greater than or equal to 2, and less than 5. This means that our solution will include all the numbers between 2 and 5, including 2 but excluding 5.

Step 2: Graph the solution

Start by drawing a number line ranged from 1 to 6. Then, plot a closed circle at 2 (since 2 is included in the solution) and an open circle at 5 (since 5 is not included in solution). Then, draw a line to connect the two circles indicating the range of the solution.

Key Concepts

Number Line

A number line is a visual representation of numbers arranged in a sequence along a straight path. It helps us to easily understand and compare the values of numbers. Here's how it typically works:

• Numbers are placed at equal intervals along the line, starting from zero at the center.
• Positive numbers are placed on the right, while negative numbers are on the left.
• A number line can extend infinitely in both directions, although we often only draw the relevant section for our purpose.

For instance, when solving inequalities, the number line becomes an invaluable tool. It helps us visualize the solution set by marking specific values and highlighting the range of numbers that are included in the solution.

Closed Circle

The concept of a closed circle on a number line is crucial when interpreting inequalities. A closed circle is used to indicate that a particular number is part of the solution set of the inequality.

• This means that if you see a closed circle on a number line, the number it is marking is included in the range of values.
• We use a closed circle when the inequality symbol is either "less than or equal to" (≤) or "greater than or equal to" (≥).

For example, in "2 ≤ x\(<5\)", we place a closed circle at 2 because 2 is included in the set of solutions. This visually confirms that x can indeed be any number starting from 2 and upwards, but excluding the next distinct boundary.

Open Circle

An open circle is another tool used on the number line to illustrate inequalities. Unlike a closed circle, it signals that a number is not part of the solution set.

• Open circles are placed at the endpoint of solutions that are strictly greater than (>) or strictly less than (<).
• The absence of filling in the circle clearly shows that the number it marks is not included in the solution set.

Considering the inequality "2 ≤ x<5", we use an open circle at 5. This tells us that while x can be any value less than 5, 5 itself is not allowed as part of the values that x can take.

Solution Set

A solution set is the collection of all possible values that satisfy an inequality. It provides a comprehensive view of the solutions, usually in both numerical and graphical form.

• Numerically, a solution set might be represented as an interval using bracket notation. For "2 ≤ x < 5", this would be written as [2, 5).
• Graphically, it is shown on the number line with closed and open circles to indicate which endpoints are included or excluded.
• It visually emphasizes the range of values that fulfill the inequality.

Understanding a solution set helps us wrap our heads around which numbers fit within the limitations of a given problem, facilitating a clearer comprehension when solving and graphing inequalities.