Problem 6 - College Algebra

Question

In Exercises 1-12, graph the solutions of each inequality on a number line. $$x \geq-5$$

Step-by-Step Solution

Answer

The solutions of the inequality \(x \geq -5\) should be graphed with a filled circle at the point representing -5 and a line stretching to the right from this point, indicating all numbers greater than or equal to -5.

Step 1: Understanding greater than or equal to inequality

The inequality \(x \geq -5\) translates to: 'x is greater than or equal to -5'. This means that x can be any number that is -5 or greater.

Step 2: Plotting the number line

Draw a number line. Mark -5 on the number line. The number -5 is included in the solution set because the inequality is 'greater than or equal to', not just 'greater than'. So, the point representing -5 on the number line should be a filled circle.

Step 3: Identifying the solution set

All numbers greater than -5 are solutions of the inequality. These are represented by all points to the right of -5 on the number line. To show this, draw a line from -5 to the right end of the number line.

Key Concepts

Number Line

The number line is a fundamental tool in mathematics that visually represents numbers as points along a straight line. It's especially useful for showing the relationships between numbers, such as which are larger or smaller, and it provides a clear way to demonstrate intervals or sets of numbers, which are essential in understanding inequalities.

To use a number line, points are marked on the line that correspond to the numbers they represent. The number line extends infinitely in both directions, with zero typically in the middle, positive numbers extending to the right, and negative numbers to the left. When graphing inequalities on a number line, it becomes a powerful visual aid to illustrate which numbers are included within the set defined by the inequality.

Greater Than or Equal To

The symbol \(\geq\) signifies an inequality known as 'greater than or equal to'. This means that when you encounter \(x \geq a\), the solution includes \(a\) and all numbers greater than \(a\). It's important not to confuse this with just \(x > a\), which would mean that \(a\) is not part of the solution.

Understanding this notation is crucial because it affects how we represent the solution on a number line. For 'greater than or equal to', we use a closed or filled circle on the number line at the point that corresponds to \(a\), indicating that \(a\) is included in the solution set. Then we shade or draw a line to the right to show all numbers greater than \(a\) are also solutions.

Solution Set of Inequality

When we talk about the solution set of an inequality, we are referring to the collection of all numbers that satisfy the inequality's condition. In the exercise \(x \geq -5\), the solution set comprises all real numbers that are greater than or equal to -5.

An excellent method to comprehend the solution set is to envisage all possible positions you could be on a number line that would make the inequality true. For our given example, anywhere from the exact point of -5 and to the right on the number line would make the inequality hold true. This forms a continuous set of numbers, showing that many, essentially infinite, values can satisfy the inequality.

Plotting Inequalities

Plotting inequalities on a number line involves a few critical steps. Firstly, you must identify the critical value in the inequality, which is -5 in the example \(x \geq -5\). Mark this value with an appropriate symbol—a filled circle since it includes the number itself.

Next, you determine the direction of the inequality. Since \(x\) is greater than or equal to -5, all numbers to the right of -5 are part of the solution. Thus, a line or shading to the right towards positive infinity signifies that every number beyond -5 on the number line satisfies the inequality. Since inequalities frequently show ranges of allowable values rather than precise answers, this visual representation is incredibly helpful for understanding them in a concrete way.