In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y = 8 - |5x + 3| and y is at least 6

Verified step by step guidance

1. Start by setting up the inequality based on the condition given: \( y = 8 - |5x + 3| \geq 6 \).
2. Rearrange the inequality to isolate the absolute value expression: \( 8 - |5x + 3| \geq 6 \) becomes \( |5x + 3| \leq 2 \).
3. Solve the inequality \( |5x + 3| \leq 2 \) by considering the two cases for absolute value: \( 5x + 3 \leq 2 \) and \( 5x + 3 \geq -2 \).
4. For the first case, solve \( 5x + 3 \leq 2 \) by subtracting 3 from both sides and then dividing by 5.
5. For the second case, solve \( 5x + 3 \geq -2 \) by subtracting 3 from both sides and then dividing by 5.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of x from zero on the number line, always yielding a non-negative result. In the context of the given equation, |5x + 3| indicates that the expression inside the absolute value can take both positive and negative values, affecting the overall equation's solutions.

Inequalities

Inequalities express a relationship where one side is not equal to the other, often using symbols like '≥' or '<'. In this problem, the condition 'y is at least 6' translates to the inequality 8 - |5x + 3| ≥ 6, which must be solved to find the range of x values that satisfy this condition.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. Understanding how to express solutions in interval notation is crucial for clearly communicating the set of x values that meet the specified conditions.