Problem 26 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$p+(-5)>-3$$

Step-by-Step Solution

Answer

The solution is \(p > 2\), and it is graphed with an open circle at 2 and a line extending to the right.

Step 1: Simplify the Inequality

The inequality given is \(p - 5 > -3\). To simplify it, you should isolate \(p\) on one side of the inequality. Start by adding 5 to both sides: \[p - 5 + 5 > -3 + 5\] This simplifies to:\[p > 2\]

Step 2: Interpret the Solution

The solution \(p > 2\) means any number greater than 2 will satisfy this inequality. There is no upper bound, so \(p\) can be any real number greater than 2.

Step 3: Graph on a Number Line

To graph \(p > 2\) on a number line, draw an open circle at 2 to indicate that 2 is not included in the solutions. Then draw a line extending to the right of 2 to show that all numbers greater than 2 are solutions. This line extends infinitely in the positive direction.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers on a straight line. It extends infinitely in both directions, with zero traditionally placed in the center. Negative numbers are found to the left of zero, while positive numbers are to the right. This line helps in visualizing mathematical concepts like inequalities, distances, and the relative position of numbers.

To work with a number line, keep these points in mind:

• Numbers increase in value as you move to the right.
• Numbers decrease in value as you move to the left.
• Open circles indicate that a particular number is not included in the solution set.
• Closed circles indicate that a number is included.

For example, when dealing with an inequality like \( p > 2 \), an open circle is used at the 2. This shows 2 is not part of the solution. The line moving to the right signifies all numbers greater than 2.

Solving Inequalities Step-by-Step

Solving inequalities is similar to solving equations, with a few additional rules to keep in mind. Here, the primary goal is to find the set of all possible values for the variable that make the inequality true.

Let's break it down:

• First, isolate the variable on one side of the inequality. This involves performing operations such as addition, subtraction, multiplication, or division.
• If you multiply or divide both sides of an inequality by a negative number, remember to flip the inequality sign. That's the key difference from solving regular equations.
• Check your work by substituting values back into the original inequality to ensure the solutions are valid.

In the example \( p + (-5) > -3 \), simplifying involves first translating it to \( p - 5 > -3 \), and then adding 5 to both sides. This leaves you with \( p > 2 \), indicating the solution includes any number greater than 2.

Graphing Inequalities Effectively

Graphing inequalities on a number line offers a visual insight into what the solution set looks like. It provides a clear method to see which values satisfy the inequality.

To graph an inequality, follow these steps:

• Determine the critical points by solving the inequality as you would an equation.
• Use open or closed circles based on whether the inequality is strict (\(>\) or \(<\)) or inclusive (\(\geq\) or \(\leq\)).
• Draw a line starting from the circle that extends towards the direction showing values that satisfy the inequality.

With \( p > 2 \), the circle at the number 2 should be open to show 2 is not part of the solution. The line going right represents all numbers greater than 2, and this extends infinitely, covering all numbers satisfying the inequality.