Problem 45 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$\frac{x-4}{6} \geq \frac{x-2}{9}+\frac{5}{18}$$

Step-by-Step Solution

Answer

The solution is \(x \geq 13\), expressed in interval notation as \([13, \infty)\). This is represented on a number line by highlighting the point at 13 and extending an arrow to the right, indicating all numbers greater than or equal to 13 are included in the solution.

Step 1: Simplify the equation

The inequality must first be simplified for ease of solution. Multiply all terms by 18 (which is the least common multiple of denominators 6, 9, and 18) in order to eliminate the fractions. The resulting inequality is \(3(x-4) \geq 2(x-2) + 5\). Further simplification gives \(3x - 12 \geq 2x - 4 + 5\).

Step 2: Simplify further and solve for x

Further simplify the inequality to \(3x -12 \geq 2x + 1\). Solving for x involves moving the 2x to the left-hand side and -12 to the right-hand side, which results in \(x \geq 13\).

Step 3: Express the solution in interval notation

The solution in interval notation form is \([13, \infty)\). This means the solution includes 13 and extends to infinity.

Step 4: Graph the solution set on a number line

On the number line, the point at 13 is colored in (or a closed dot is placed at 13) indicating that the number 13 is included in the solution set. An arrow is drawn extending to the right of 13, towards positive infinity to represent all the numbers greater than 13 that are part of the solution set.