Math Problem Statement
Solution
The given problem asks to find the solution of the system of inequalities:
4(x - 3) - 3(x - 2) < 5, \\ -6x + 7 \leq 0. \end{cases}$$ Let's solve these step by step. --- ### 1st Inequality: $$4(x - 3) - 3(x - 2) < 5$$ Expand both terms: $$4x - 12 - 3x + 6 < 5$$ Combine like terms: $$x - 6 < 5$$ Solve for $$x$$: $$x < 11$$ --- ### 2nd Inequality: $$-6x + 7 \leq 0$$ Rearrange: $$-6x \leq -7$$ Divide by $$-6$$ (reverse the inequality because dividing by a negative): $$x \geq \frac{7}{6}$$ --- ### Combine the Solutions: From the first inequality, $$x < 11$$. From the second inequality, $$x \geq \frac{7}{6}$$. Thus, the solution is: $$\frac{7}{6} \leq x < 11$$ In interval notation: $$\left[\frac{7}{6}, 11\right)$$ --- From the options in the image, the correct one is **$$\left[\frac{7}{6}, 11\right)$$**. Would you like me to explain further or solve a related problem? --- ### Related Questions: 1. How do you handle strict inequalities when combining solutions? 2. What happens if there are no overlapping intervals in a system of inequalities? 3. Can you explain when and why inequalities reverse when multiplying or dividing by a negative number? 4. How do you graph the solution of a system of inequalities on a number line? 5. What is the difference between "and" and "or" in systems of inequalities? ### Tip: When solving a system of inequalities, always check the intersection (overlap) of the solutions to ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Systems of Inequalities
Interval Notation
Formulas
Distributive property: a(b + c) = ab + ac
Rules for solving linear inequalities: adding, subtracting, multiplying, dividing
Intersection of solutions for systems of inequalities
Theorems
Property of inequalities: When multiplying or dividing by a negative number, reverse the inequality sign
Suitable Grade Level
Grades 8-10
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