Solve the inequality and specify the answer using interval notation. (a) 3 x+5<17 (b) |3 x+5|<17

step 1 I'm going to solve the inequality, so we're going to start by isolating our variables, so we're

Solve the inequality and specify the answer using interval...

Key Concepts

Absolute Value Inequality

Absolute value inequalities involve expressions within absolute value symbols and represent distances on the number line. They can be categorized into two types: those in the form |expression| < constant and |expression| > constant. The inequality |expression| < constant is equivalent to a compound inequality -constant < expression < constant, whereas |expression| > constant results in two parts: expression < -constant or expression > constant. This reflects whether all solutions that are within a certain distance (less than) or outside of that distance (greater than) from zero are considered.

Linear Inequality

A linear inequality involves a linear expression on one or both sides of the inequality symbol. The process of solving it is similar to solving a linear equation: isolate the variable by performing algebraic operations such as addition, subtraction, multiplication, or division. During the process, one must be cautious when multiplying or dividing by a negative number, as this reverses the direction of the inequality sign.

Interval Notation

Interval notation is a concise way to represent sets of numbers along the number line. It uses parentheses and brackets to denote whether endpoints are excluded or included, respectively. Open intervals (using parentheses) indicate endpoints are not part of the solution, while closed intervals (using brackets) indicate that endpoints are included. This notation is especially useful for expressing the solution sets of inequalities.

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