Problem 66 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|-0.5 x+5| \geq 4$$

Step-by-Step Solution

Answer

The solution is \((- \infty, 2] \cup [18, \infty)\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|-0.5x + 5| \geq 4\) indicates that the expression inside the absolute value can be either greater than or equal to 4, or less than or equal to -4.

Step 2: Separate into Two Inequalities

Rewrite the inequality as two separate inequalities without the absolute value: 1. \(-0.5x + 5 \geq 4\)2. \(-0.5x + 5 \leq -4\).

Step 3: Solve the First Inequality

Solve the inequality \(-0.5x + 5 \geq 4\):Subtract 5 from both sides: \(-0.5x \geq -1\)Divide both sides by -0.5 (remember the inequality sign flips when dividing by a negative number):\(x \leq 2\).

Step 4: Solve the Second Inequality

Solve the inequality \(-0.5x + 5 \leq -4\):Subtract 5 from both sides:\(-0.5x \leq -9\)Divide both sides by -0.5 (the inequality sign flips again):\(x \geq 18\).

Step 5: Combine Solutions

Combine the solutions \(x \leq 2\) and \(x \geq 18\) into the interval notation. The solution in interval notation is:\((-\infty, 2] \cup [18, \infty)\). This is because the values of \(x\) satisfying the inequality are those less than or equal to 2 and greater than or equal to 18.

Key Concepts

Interval Notation

Interval notation is a concise way to express inequality solutions, providing a clear and formatted way to represent ranges of numbers on a number line. This method is particularly useful in algebraic contexts, allowing us to easily communicate solutions to inequalities.
When we write an interval:

• A square bracket \([\text{ or } ]\) indicates that the endpoint is included in the interval, i.e., the number is part of the solution set.
• A parenthesis \((\text{ or } )\) indicates that the endpoint is not included, representing solutions that get infinitely close to but do not actually include the boundary number.

In the case of the inequality \(|-0.5x + 5| \geq 4\), once solved, the solution in interval notation is \((\-\infty, 2] \cup [18, \infty)\). This means that all numbers from negative infinity up to 2 (inclusive) and all numbers from 18 (inclusive) to positive infinity satisfy the inequality.

Inequality Solutions

Solving inequalities is about finding the range of values for which the inequality holds true. For inequalities involving absolute values like \(|-0.5x + 5| \geq 4\), we need to consider two cases because absolute values can represent both positive and negative scenarios.
The original equation can be split into two separate inequalities:

• \(-0.5x + 5 \geq 4\)
• \(-0.5x + 5 \leq -4\)

When solving inequality problems, it's critical to remember that the direction of the inequality sign reverses when we multiply or divide by a negative number.
By solving each inequality separately, we then find the legitimate values of \(x\). Once each inequality is solved, you can combine the solutions to form a comprehensive answer in interval notation, which neatly displays the range of valid numbers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators that represent values. In the inequality \(|-0.5x + 5| \geq 4\), the expression \(-0.5x + 5\) is held within an absolute value symbol, representing how far this expression's value is from zero.
Understanding and manipulating algebraic expressions is crucial when dealing with inequalities as it allows you to isolate variables and determine their possible values. Here are some key points:

• When you encounter absolute value signs, it indicates that the expression inside can have a dual nature: one being positive and one negative.
• Remember to handle the expressions carefully, keeping in mind operations that change the direction of inequalities, such as multiplying or dividing by a negative number.

Grasping these principles helps in solving absolute value inequalities by providing a structured approach to dissect and handle the problem step-by-step.