In Exercises 59–94, solve each absolute value inequality. 1 < |2 - 3x|

Verified step by step guidance

1. Start by understanding the absolute value inequality: $1<|2-3x|$. This means the expression inside the absolute value, $2-3x$, is either greater than 1 or less than -1.
2. Split the inequality into two separate inequalities: $2-3x>1$ and $2-3x<-1$.
3. Solve the first inequality: $2-3x>1$. Subtract 2 from both sides to get $-3x>-1$. Then, divide both sides by -3, remembering to reverse the inequality sign, resulting in $x<\frac{1}{3}$.
4. Solve the second inequality: $2-3x<-1$. Subtract 2 from both sides to get $-3x<-3$. Then, divide both sides by -3, remembering to reverse the inequality sign, resulting in $x>1$.
5. Combine the solutions from both inequalities. The solution to the original inequality is $x<\frac{1}{3}$ or $x>1$.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving absolute value inequalities, it is important to consider the two cases that arise from the definition of absolute value, leading to two separate inequalities to solve.

Solving Absolute Value Inequalities

To solve an absolute value inequality like 1 < |2 - 3x|, we break it into two cases based on the definition of absolute value. This results in two separate inequalities: 2 - 3x > 1 and 2 - 3x < -1. Each inequality is then solved independently, and the solutions are combined to find the overall solution set.