Problem 36 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$8 x+3>3(2 x+1)+x+5$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 5\), which can be expressed as \((5, \infty)\) in interval notation. The graph is a number line with an open-circle at 5 and an arrow pointing to the right, indicating all values greater than 5.

Step 1: Simplify the right side of the inequality

Firstly, distribute the 3 on the right side of the inequality: 3(2x + 1) turns into 6x + 3. Then, add the \(x\) and 5 to get 6x + 3 + x + 5, which simplifies to 7x + 8.

Step 2: Combine like terms

Replace the right side of the inequality with the simplified expression. So, the inequality changes from \(8x + 3 > 3(2x + 1) + x + 5\) to \(8x + 3 > 7x + 8\). Now, get all terms with \(x\) on one side by subtracting 7x from both sides to get \(x + 3 > 8\).

Step 3: Formulate the solution

Subtract 3 from both sides to isolate \(x\), bringing the inequality to \(x > 5\). This is the solution to the inequality in algebraic form.

Step 4: Express the solution in interval notation

Since \(x > 5\), the solution is all numbers greater than 5. The interval notation for this is \((5, \infty)\), where parentheses mean the endpoint is not included.

Step 5: Graph the solution set on a number line

Draw a number line, and mark the point 5. Draw a circle at 5 which is open to indicate that it is not part of the solution, and draw an arrow to the right from 5 towards greater values.