Math Problem Statement
Solve the system of inequalities: 2(3 - 2x) > 8 - 5x and 10 - x > 2.
Solution
The image contains a system of inequalities:
2(3 - 2x) > 8 - 5x, \\ 10 - x > 2. \end{cases}$$ ### Let's solve each inequality step by step: #### First Inequality: $$2(3 - 2x) > 8 - 5x$$ 1. Distribute the $$2$$ on the left side: $$6 - 4x > 8 - 5x$$ 2. Add $$5x$$ to both sides to get all terms involving $$x$$ on one side: $$6 + x > 8$$ 3. Subtract $$6$$ from both sides: $$x > 2$$ #### Second Inequality: $$10 - x > 2$$ 1. Subtract $$10$$ from both sides: $$-x > -8$$ 2. Multiply both sides by $$-1$$ (and remember to flip the inequality sign): $$x < 8$$ ### Final Answer: The solution to the system of inequalities is: $$2 < x < 8$$ This means $$x$$ is in the interval $$(2, 8)$$. Would you like further details, or do you have any questions? ### Related Questions: 1. How would the solution change if one inequality was $$\geq$$ instead of $$>$$? 2. What happens if both inequalities had the sign $$\leq$$? 3. How can you graph the solution set on a number line? 4. Can you solve a similar system of inequalities with a quadratic inequality involved? 5. What is the impact of multiplying an inequality by a negative number? ### Tip: When solving inequalities, always remember to flip the inequality sign if you multiply or divide both sides by a negative number.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Inequalities
Algebra
Linear Equations
Formulas
Distributive Property
Solving Linear Inequalities
Theorems
Basic Rules of Inequalities
Suitable Grade Level
Grades 7-9