Problem 34 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$18 x+45 \leq 12 x-8$$

Step-by-Step Solution

Answer

The solution to the inequality \(18x + 45 \leq 12x -8\) is \( x \leq -53/6\), expressed in interval notation as \([-∞, -53/6]\). The graph would start at -53/6 with a closed circle and shade to the left.

Step 1: Simplify the Inequality

Subtract 12x from both sides to isolate the x variable. This will result in: \(6x + 45 \leq -8\)

Step 2: Further Isolate x

In order to completely isolate x, it's important to get rid of '+45' on its side. Therefore, subtract 45 from both sides. This results in: \(6x \leq -53\)

Step 3: Solving for x

Now that x is isolated on one side, divide both sides by 6 to find the final solution: \(x \leq -53/6\) or in decimal form: \(x \leq -8.83\)

Step 4: Express in Interval Notation

The values x can take on are less than or equal to -53/6. This is expressed in interval notation as: \([-∞, -53/6]\)

Step 5: Graph on a Number Line

Draw a number line and mark -53/6. Draw a closed circle (indicating 'less than or equal to') at this point and shade to the left (indicating all numbers less than -53/6).