Problem 30 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$8 x-2 \geq 14$$

Step-by-Step Solution

Answer

The solution to the inequality \(8x - 2 \geq 14\) is \(x \geq 2\), which in interval notation is represented as \([2, \infty)\). The solution set can be graphed on a number line with a filled dot at 2 and an arrow pointing towards +∞.

Step 1: Solve the inequality

To solve the inequality \(8x - 2 \geq 14\), first, balance the inequality by adding 2 to both sides to isolate the term with x. This results in \(8x \geq 16\). Next, divide both sides by 8 to solve for x, getting the result \(x \geq 2\)

Step 2: Write the solution in interval notation

Interval notation is a way of writing subsets of the real number line. In interval notation, the solution \(x \geq 2\) is represented as \([2, \infty)\). The square bracket [ means the interval includes 2, and the parenthesis ) after ∞ means it extends indefinitely but doesn't include ∞ as ∞ is not a real number.

Step 3: Graph the solution on a number line

To graph the solution on a number line, draw a line and mark out the point corresponding to 2. Since the solution includes 2, place a filled dot at 2. As the solution set extends indefinitely towards +∞, draw an arrow starting from 2 and pointing towards the positive direction of the number line.