Problem 29 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$3 x-7 \geq 13$$

Step-by-Step Solution

Answer

The solution to the inequality \(3x - 7 \geq 13\) is \(x \geq \frac{20}{3}\). In interval notation, this solution is written as \([\frac{20}{3}, \infty)\). On the number line, the solution set includes the point \(\frac{20}{3}\) and stretches towards infinity to the right from this point.

Step 1: Solving the Inequality

Start by isolating the variable \(x\). This can be done by performing the same mathematical operations on both sides of the inequality until \(x\) is by itself on one side. \n In this case, add 7 to both sides of the inequality to get rid of the -7 on the left side. The inequality now looks like this: \(3x \geq 20\). Then, divide both sides by 3 to solve for \(x\). We get \(x \geq \frac{20}{3}\).

Step 2: Applying Interval Notation

Interval notation is a simplified form of writing the solution to an inequality. In this case, \(x \geq \frac{20}{3}\) means that \(x\) can be any number that is greater than or equal to \(\frac{20}{3}\). In interval notation, this is written as \([\frac{20}{3}, \infty)\). The square bracket indicates that \(\frac{20}{3}\) is included in the solution set, and the parenthesis implies that infinity is not a reachable number but the values of \(x\) tend to infinity.

Step 3: Graphing the Solution Set

To graph this solution set on a number line, one starts by drawing a line. The point corresponding to \(\frac{20}{3}\) needs to be marked with a solid dot (which signifies that the number is included in the solution set) and an arrow pointing towards the right (to signify that all numbers greater than this point are part of the solution set).