Math Problem Statement
Solve each of the given compound inequalities. Enter your answers using interval notation.
-7x - 1 < 41 or 2x + 3 ≥ -1
-7x - 1 < 41 and 2x + 3 ≥ -1
Solution
Let's solve both compound inequalities step by step.
1. First Compound Inequality:
First part:
x > -6 \quad \text{(divide by -7 and flip the inequality sign)}$$ #### Second part: $$2x + 3 \geq -1$$ $$2x \geq -4 \quad \text{(subtract 3 from both sides)} \\ x \geq -2 \quad \text{(divide by 2)}$$ **Solution for the "or" inequality:** The solution is the union of these two inequalities: $$x > -6$$ or $$x \geq -2$$. This simplifies to: $$\boxed{(-6, \infty)}$$ --- ### 2. **Second Compound Inequality:** $$-7x - 1 < 41 \quad \text{and} \quad 2x + 3 \geq -1$$ #### First part: $$-7x - 1 < 41$$ (Same as above) $$x > -6$$ #### Second part: $$2x + 3 \geq -1$$ (Same as above) $$x \geq -2$$ **Solution for the "and" inequality:** The solution is the intersection of these two inequalities: $$x > -6$$ and $$x \geq -2$$. This simplifies to: $$\boxed{[-2, \infty)}$$ --- ### Final Solutions: 1. For the "or" inequality: $$(-6, \infty)$$ 2. For the "and" inequality: $$[-2, \infty)$$ Let me know if you need detailed interval illustrations or further clarifications! Here are some related questions: 1. What is the difference between "and" and "or" in compound inequalities? 2. How do you interpret interval notation? 3. How does flipping the inequality sign work when dividing by a negative number? 4. Can a compound inequality have no solution? 5. Why does $$(-6, \infty)$$ encompass $$[-2, \infty)$$ in this case? **Tip:** Always solve each part of a compound inequality separately before combining the results with union or intersection!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Compound Inequalities
Interval Notation
Algebraic Manipulation
Formulas
Inequality operations: add, subtract, multiply, divide (flip inequality for division by negatives)
Intersection (AND) and Union (OR) of solution sets
Theorems
Properties of Inequalities
Set Theory for Intervals
Suitable Grade Level
Grades 8-10