Problem 27 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$5 x+11<26$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 3\), graphically represented on a number line with an arrow pointing to the left from 3. The interval notation for the solution set is \((-\infty, 3)\).

Step 1: Solving the Inequality

Start by subtracting 11 from both sides of the inequality to isolate the term containing x. This yields \(5x < 26 - 11\), which simplifies to \(5x < 15\). Dividing both sides of the inequality by 5 results in \(x < 3\).

Step 2: Graphing the Solution on a Number Line

Arrange the number line with a scale appropriate for the solution (in this case, from 0 to 5 would be adequate). Place a hollow circle above the number 3 on the number line, indicating that 3 is not included in the solution set, and draw an arrow to the left of 3, showing that all numbers less than 3 are part of the solution set.

Step 3: Expressing the Solution in Interval Notation

The last step is to express the solution in interval notation, which is a concise way to represent a set of numbers. As the solution contains all numbers less than 3, but not 3 itself, the interval notation is \((-\infty, 3)\), where \(-\infty\) represents negative infinity and the parentheses indicate that the endpoints are not included in the solution.

Key Concepts

Interval Notation

When expressing the range of solutions for an inequality, interval notation provides a compact form to convey all the numbers that make up the solution. To grasp this concept, imagine a continuous stretch of numbers along a line. The interval notation for the solution of the inequality 5x + 11 < 26 is written as \( (-\infty, 3) \). This tells us that the solution includes all numbers from negative infinity up to, but not including, the number 3. The use of parentheses \( ( ) \) signifies that the endpoint, in this case 3, is not part of the solution set. If it were included, we would use a bracket \([ [ \), instead. Interval notation is efficient and eliminates ambiguity, making it essential for clearly communicating solutions to inequalities.

Graphing on a Number Line

Visualizing the solution to an inequality on a number line helps students quickly see the range of possible values that satisfy the inequality. For the inequality x < 3, which originates from solving \(5x + 11 < 26\), you would draw a number line. You then place a hollow circle, or open dot, above the number 3 to indicate that 3 is not included in the solution set. A solid dot would mean the number is included. From the open circle at 3, you extend an arrow to the left to represent all numbers less than 3. This clear visualization aids in understanding the set of solutions and serves as a handy reference for checking answers.

Inequality Solutions

Understanding solutions to inequalities is critical in mathematics. It's about determining which values, when substituted into an inequality, make the statement true. For the linear inequality 5x + 11 < 26, we solve for x by isolating the variable on one side. This results in the simplified inequality \(x < 3\). The solution encompasses every number that is less than 3, but does not include 3 itself. It's this set of numbers that makes the original inequality true. Confidently solving and graphing these solutions arms students with the skills to tackle various applications in algebra, including modeling and solving real-world problems.