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Chapter 0: Problem 96

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=8-|5 x+3|\) and \(y\) is at least 6

Short Answer

Expert verified
The solution to this problem in interval notation is \((-鈭�, -1]\) and \([-0.2, 鈭�)\).

Step by step solution

01

Understand the Conditions

Firstly, consider the conditions provided. \(y\) is at least 6, it means \(y \geq 6\). So, we can set up an inequality based on the given conditions: \(8 - |5x + 3| \geq 6\).
02

Simplify the Inequality Expression

Rewrite the inequality by subtracting the constant on both sides: \(- |5x + 3| \geq -2\).
03

Solving the Absolute Value Term

Since -2 is a negative number, we can express the absolute value inequality expression as a set containing two inequalities: \(5x + 3 \leq -2\) and \(5x + 3 \geq 2\). Using these, we solve for \(x\).
04

Solve the Two Inequalities Separately

Solving the inequalities independently: \(5x + 3 \leq -2\) simplifies to \(x \leq -1\). And \(5x + 3 \geq 2\) simplifies to \(x \geq -0.2\).
05

Express the Solution using Interval Notation

The interval notation for the solution is \((-鈭�, -1]\) for \(x\leq -1\) and \([-0.2, 鈭�)\) for \(x\geq -0.2\).

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Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

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When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

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