Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(3 x-7>2\)

Short answer

\(x > 3\) in interval notation is \((3, ∞)\).

Step-by-step solution

Isolate the Variable

To isolate the variable, add 7 to both sides of the inequality:\(3x - 7 + 7 > 2 + 7\)This simplifies to:\(3x > 9\)

Solve for the Variable

To solve for \(x\), divide both sides of the inequality by 3:\(\frac{3x}{3} > \frac{9}{3}\)This simplifies to:\(x > 3\)

Express the Solution in Interval Notation

The solution can be expressed in interval notation as:\((3, ∞)\)

Graph the Solution Set

On the number line, an open circle is placed at 3 and a ray is drawn to the right, indicating all numbers greater than 3.

Key concepts

Isolate the Variable

When solving inequalities, the first step is to isolate the variable on one side of the inequality. This is similar to solving equations, but you must be cautious of the inequality sign. For the inequality given,
you start with:\(3x - 7 > 2\)To isolate the variable, you need to get rid of the -7 on the left-hand side. You do this by adding 7 to both sides of the inequality.The equation then looks like this:\(3x - 7 + 7 > 2 + 7\)Simplify both sides to make it more manageable:\(3x > 9\)Now the variable x is isolated, and you can proceed to solving the inequality.

Interval Notation

After isolating the variable and solving for it, you must express the solution using interval notation. This notation helps represent the set of all possible solutions in a concise way. In our example, we have isolated x and found that:\(x > 3\)In interval notation, this is written as:\( (3, ∞) \)Here's why:

• Parentheses \((\) are used to indicate that the endpoint is not included in the solution. An open circle on the graph also denotes this.
• The symbol ∞ (infinity) is used because there is no upper limit to the values that x can take.

So, \((3, ∞)\) signifies all the numbers greater than 3 but not including 3 itself.

Graphing Inequalities

Graphing inequalities helps visually interpret the solution set. To graph our solution from the earlier steps, follow these steps:

• Draw a number line.
• Place an open circle at 3 to indicate that 3 is not included in the solution.
• Draw a ray starting from the open circle at 3 and extending to the right, showing all numbers greater than 3 are part of the solution set.

The open circle is crucial because it differentiates between numbers that are part of the solution and those that are not. Using this method, you can visually capture the essence of the inequality:\(3x - 7 > 2\) simplifies to \(x > 3\), and its graphical representation will help you and others understand the solution more effectively.