Problem 29 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |x|>3 $$

Step-by-Step Solution

Answer

The solution is \(k < -7\) or \((-\infty, -7)\).

Step 1: Subtract 1 from both sides

To isolate the term with the variable, subtract 1 from both sides of the inequality:\[3k + 1 - 1 < -20 - 1\]This simplifies to:\[3k < -21\]

Step 2: Divide both sides by 3

Next, divide both sides of the inequality by 3 to solve for the variable k:\[\frac{3k}{3} < \frac{-21}{3}\]This simplifies to:\[k < -7\]

Step 3: Graph the solution set

Graph the inequality by drawing a number line. Plot an open circle at -7 (since -7 is not included in the solution) and shade all the numbers to the left of -7 to indicate that all those values are part of the solution set.

Step 4: Write the solution in interval notation

Write the solution as an interval where all numbers less than -7 are included. The interval notation for this solution is:\((-\infty, -7)\)

Key Concepts

inequality solutions

Solving inequalities is a lot like solving equations, except for one key difference: the direction of the inequality matters. The goal is to isolate the variable on one side of the inequality symbol. For instance, starting with the inequality 3k + 1 < -20:

• First, we subtract 1 from both sides to isolate the term involving the variable.
• This results in the new inequality 3k < -21.
• Next, we divide by 3 on both sides to solve for k.

This simplification leads us to k < -7. The inequality solution tells us that k can be any number less than -7.

graphing inequalities

Once you've solved an inequality, the next step is to graph the solution set. This helps you visualize all the possible values the variable can take. To graph the solution k < -7:

• Draw a number line.
• Place an open circle at -7 because -7 is not included in the solution (if it were, you'd use a closed circle instead).
• Shade the line to the left of -7 to indicate all numbers less than -7 are part of the solution set.

Graphing inequalities provides a clear, visual representation of the range of possible values for the variable.

interval notation

Interval notation is a concise way to describe the range of values that satisfy an inequality. Here’s how it works:

• An open parenthesis '(' or ')' indicates that the endpoint is not included.
• A closed bracket '[' or ']' indicates that the endpoint is included.

For the inequality k < -7, since -7 is not included, we use an open parenthesis. We also include '-∞' because the solution extends infinitely to the left, so the interval notation is (-∞, -7). This notation efficiently communicates all values in the solution set without needing to graph.