Problem 27 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$5 x+11<26$$

Step-by-Step Solution

Answer

The solution to the inequality \(5x+11<26\) is \( x < 3 \), which in interval notation is expressed as \((-\infty, 3)\).

Step 1: Simplify inequality

Start by isolating the term with \( x \) on one side of the inequality. This is done by subtracting 11 from both sides of the inequality: \(5x + 11 - 11 < 26 - 11\), which simplifies to \(5x < 15\)

Step 2: Solve for x

To solve for \( x \), divide both sides of the inequality by 5. \( \frac{5x}{5} < \frac{15}{5} \), simplifies to \( x < 3 \)

Step 3: Express in Interval Notation

Our solution \( x < 3 \) denotes that \( x \) can be any number less than 3 but not 3 itself. In interval notation, it's expressed as \((-\infty, 3)\)

Step 4: Graph the solution on a Number Line

Draw a number line, and mark the number 3. Then draw an open circle at 3 (Because 3 is not part of the solution set), and shade all the values to the left of 3, extending the line to indicate that it continues towards negative infinity.