In Exercises 114-117 , solve the inequality and sketch the solution on the real number line. Use

step 1 This question will use the inequality of the form absolute of x plus 8 minus 1 greater equal to

In Exercises 114-117 , solve the inequality and sketch the...

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Key Concepts

Absolute Value Functions

Absolute value functions take any real number and output its magnitude, removing any negative sign. They are inherently nonnegative and involve case distinctions based on the sign of the input expression, which is crucial for understanding and solving equations and inequalities that include absolute values.

Inequalities Involving Absolute Values

Inequalities with absolute value expressions typically require splitting the problem into separate cases depending on the sign of the expression inside the absolute value. This process helps to convert the absolute value inequality into two linear inequalities, which can then be solved using standard techniques.

Graphing Solutions on the Number Line

After solving an inequality, the solution set can be depicted on the real number line by marking key boundary points and shading intervals that satisfy the inequality. This visual representation aids in understanding the extent and nature of the solution set.

Using Graphing Utilities for Verification

Graphing utilities offer a way to visually verify the solutions obtained algebraically. By graphing the functions involved and the inequality conditions, one can confirm areas of intersection or regions where the inequality holds, adding confidence to the solution process.

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