Solve each equation and inequality. For the inequalities, graph the solution set and write it us

step 1 Got an absolute value inequality here. The opposite of the absolute value of 3x plus 1 is less t

Solve each equation and inequality. For the inequalities,...

Key Concepts

Absolute Value

Absolute value represents the non-negative distance of a number or expression from zero on the number line. It is defined such that for any real number x, |x| equals x if x is non-negative and -x if x is negative. This concept is fundamental when dealing with expressions where only magnitude matters regardless of sign.

Absolute Value Inequalities

Absolute value inequalities involve determining the set of values for which the absolute value of an expression satisfies a given inequality condition. These are typically handled by considering separate cases based on the definition of absolute value. For example, inequalities of the form |expression| > constant are split into two cases: one where the expression is greater than the constant and another where it is less than the negative of the constant.

Inequality Manipulation and Reversal

When solving inequalities, especially when multiplying or dividing both sides by a negative number, it is crucial to reverse the inequality sign. This property maintains the correctness of the inequality and is a key step in isolating the variable during the problem-solving process.

Graphing and Interval Notation

Graphing the solution set on a number line provides a visual representation of the values that satisfy an inequality. Interval notation is a concise way to express these solution sets by using brackets and parentheses to indicate the inclusion or exclusion of endpoints. Both methods are integral in accurately communicating the results of an inequality problem.

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