Question

In Problems 31-34, graph the numbers \(x\) on the real number line. \(x>-1\)

Short answer

Draw a number line, place an open circle at -1, and shade everything to the right of -1.

Step-by-step solution

Identify the Inequality

The inequality given is \(x > -1\). This means we need to find all values of \(x\) that are greater than -1.

Determine the Critical Point

The critical point here is -1. This is the point where the inequality changes. However, since the inequality is strictly greater than (-1), -1 is not included.

Plot the Critical Point on the Number Line

Draw a number line and locate the point \(-1\). Represent this point with an open circle to indicate that -1 is not included in the solution.

Shade the Region

Shade the region to the right of the open circle. This shaded region represents all numbers greater than -1 on the real number line.

Key concepts

Number line

The number line is a visual representation of all real numbers ordered from left to right. It's like a ruler without units, where each point corresponds to a number. To draw a number line for our problem, you simply draw a horizontal line and mark relevant points. For instance, in our exercise, we need to find out where -1 is and where other nearby numbers such as -2, 0, and 1 are. This will help you accurately place and understand the numbers you are dealing with.

Inequality

An inequality is a mathematical expression that shows the relationship between two values. Instead of using an equal sign, it uses signs like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). In our particular problem, we have the inequality x > -1. This means that x can take any value that is larger than -1. Note that -1 itself is not included because the sign is 'greater than' and not 'greater than or equal to'.

Critical point

The critical point in an inequality is the value that divides the number line into parts that satisfy the inequality and parts that do not. For x > -1, the critical point is -1. It's essential to mark this clearly on the number line. Since our inequality does not include -1 (it's a 'greater than' and not 'greater than or equal to'), we represent -1 with an open circle. This open circle shows that -1 itself is not a part of the solutions but forms the border for all solutions greater than itself.

Graphing technique

Graphing an inequality on a number line is straightforward once you understand the steps. First, identify the critical point and mark it with either an open or closed circle. In our example, because x > -1, we use an open circle. Next, shade the line to the right of -1. This shading represents all the numbers greater than -1. The shaded region visually shows the solution set for the inequality. Whenever you deal with inequalities, use this graphing technique to help you easily understand and communicate the solution.