Use interval notation to represent all values of x satisfying the given conditions. y = |2x - 5| + 1 and y > 9

Verified step by step guidance

1. Start by isolating the absolute value expression in the inequality: given \( y = |2x - 5| + 1 \) and \( y > 9 \), substitute \( y \) to get \( |2x - 5| + 1 > 9 \).
2. Subtract 1 from both sides to simplify the inequality: \( |2x - 5| > 8 \).
3. Recall that for an absolute value inequality \( |A| > B \) where \( B > 0 \), the solution splits into two cases: \( A > B \) or \( A < -B \). Apply this to get two inequalities: \( 2x - 5 > 8 \) or \( 2x - 5 < -8 \).
4. Solve each inequality separately: - For \( 2x - 5 > 8 \), add 5 to both sides to get \( 2x > 13 \), then divide by 2 to find \( x > \frac{13}{2} \). - For \( 2x - 5 < -8 \), add 5 to both sides to get \( 2x < -3 \), then divide by 2 to find \( x < -\frac{3}{2} \).
5. Express the solution in interval notation by combining both parts: \( (-\infty, -\frac{3}{2}) \cup (\frac{13}{2}, \infty) \).

Key Concepts

Absolute Value Functions

An absolute value function outputs the distance of a number from zero, always yielding a non-negative result. For example, |2x - 5| represents the distance of the expression (2x - 5) from zero, affecting how the function behaves on either side of the point where 2x - 5 equals zero.

Inequalities Involving Absolute Values

Solving inequalities with absolute values requires considering two cases: one where the expression inside the absolute value is positive and one where it is negative. For y > 9 with y = |2x - 5| + 1, you isolate the absolute value and solve the resulting inequalities separately to find the solution set.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints). This notation is essential for expressing the solution set of inequalities clearly and efficiently.