Question

Solve the inequality and graph its solution. \(x-3>2\)

Short answer

The solution to the inequality is \(x > 5\). This is represented on a number line with an open circle at 5 and an arrow pointing to the right.

Step-by-step solution

Isolate the variable

We start by isolating the variable \(x\). To do this, we add 3 to both sides of the inequality, which gives us \(x > 2 + 3\).

Simplify the expression

Next, we simplify the expression on the right side of the inequality to get the actual value of \(x\), which gives us \(x > 5\).

Graph the solution

The solution \(x > 5\) means that \(x\) can be any number greater than 5. To represent this on a number line, we place an open circle at 5 (because 5 is not included in the solution) and draw an arrow pointing to the right (representing all numbers greater than 5).

Key concepts

Inequality Graphing

When studying inequalities, understanding how to represent them visually is essential. Inequality graphing is a technique used to visually demonstrate the range of possible solutions to an inequality. Let's consider the inequality from the exercise, where we found the solution to be \(x > 5\). To graph this inequality, we use a number line.

Firstly, we locate the number 5 on the number line. Since our inequality does not include the number 5 itself (it is strictly greater than 5), we use an open circle to indicate that 5 is not a part of the solution set. Next, since \(x\) can be any number greater than 5, we draw a ray (an arrow) starting at 5 and extending to the right, showcasing that all numbers to the right of 5 satisfy the inequality. This visual representation is intuitive and helps students quickly identify the set of possible solutions.

Isolating Variables

The process of isolating variables is crucial when solving inequalities, just as it is with equations. It involves manipulating the inequality to get the variable of interest by itself on one side of the inequality sign. In our exercise, we isolated \(x\) by adding 3 to both sides of the inequality \(x - 3 > 2\), obtaining \(x > 5\).

To do this effectively, one must follow the same mathematical operations on both sides of the inequality, respecting the rules of algebra. These steps might include addition, subtraction, multiplication, and division. However, it's important to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. Isolating variables is a fundamental skill that allows students to identify the solution set and proceed towards graphing or further analysis.

Simplifying Expressions

Simplifying expressions is a key aspect of solving inequalities and can be thought of as cleaning up the mathematical statement to make it easier to understand and solve. In our example, after adding 3 to both sides, we need to combine the constant terms to simplify the expression. We merge 2 and 3 to get a single number, \(5\), simplifying the inequality to \(x > 5\).

This step removes any unnecessary complexity and allows us to clearly see the critical information: the threshold number that \(x\) needs to be greater than. Simplifying expressions often involves combining like terms, reducing fractions, or performing other algebraic manipulations to achieve the most straightforward form. Teaching students how to simplify effectively equips them with a powerful tool to navigate more complex inequalities and algebraic expressions in general.