Question - Solving Absolute Value Inequalities
Solution:
To solve the inequality $$ |c| - 1 \geq 11 $$, let's first isolate the absolute value expression on one side.$$ |c| - 1 \geq 11 $$Add 1 to both sides to get:$$ |c| \geq 12 $$This inequality means that the distance of $$ c $$ from 0 on the number line is greater than or equal to 12. Absolute value inequalities like this one split into two cases because the variable inside the absolute value can be either positive or negative.Case 1: If $$ c $$ is positive or zero, then we remove the absolute value:$$ c \geq 12 $$Case 2: If $$ c $$ is negative, the absolute value makes it positive, so we have:$$ -c \geq 12 $$Multiply both sides by -1, remembering to reverse the inequality sign since multiplying an inequality by a negative number reverses the direction of the inequality:$$ c \leq -12 $$Now, we combine both cases into a compound inequality:$$ c \leq -12 $$ or $$ c \geq 12 $$These inequalities represent all the possible values of $$ c $$ that make the original inequality true. In interval notation, this solution would be:$$ (-\infty, -12] \cup [12, \infty) $$
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