Problem 28 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$2 x+5<17$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 6\), which is expressed in interval notation as \((-\infty, 6)\).

Step 1: Isolate the Variable

First, subtract 5 from both sides of the inequality to facilitate isolating the variable on one side. This gives \(2x < 17 - 5\), which simplifies to \(2x < 12\).

Step 2: Solve for \(x\)

Next, divide each side of the inequality by 2. This gives \(x < 12 / 2\), which simplifies to \(x < 6\). This means that the solution to the inequality includes all numbers less than 6.

Step 3: Express Solution in Interval Notation

The solution, \(x < 6\), can be translated into interval notation as \((-\infty, 6)\). This interval represents all real numbers less than 6.

Step 4: Graph the Solution

To graph this on a number line, place a circle on 6 (because 6 is not included in the solution set). Draw a line from the circle extending left towards negative infinity, indicating that all numbers less than 6 are part of the solution set.