Math Problem Statement
Solve the given system of inequalities from the handwritten mathematical problem.
Solution
The image contains a mathematical task asking to solve a system of inequalities. Let's break it down step by step:
1. First system of inequalities:
2 - \frac{3 + 2x}{3} > 7 - \frac{x + 6}{2} \\ 3 - \frac{x}{4} < x \end{cases}$$ #### Step 1.1: Solve the first inequality $$2 - \frac{3 + 2x}{3} > 7 - \frac{x + 6}{2}$$ 1. Simplify fractions: $$\frac{3 + 2x}{3} = \frac{3}{3} + \frac{2x}{3} = 1 + \frac{2x}{3}, \quad \frac{x + 6}{2} = \frac{x}{2} + 3$$ Rewrite the inequality: $$2 - \left(1 + \frac{2x}{3}\right) > 7 - \left(\frac{x}{2} + 3\right)$$ 2. Simplify further: $$2 - 1 - \frac{2x}{3} > 7 - 3 - \frac{x}{2}$$ $$1 - \frac{2x}{3} > 4 - \frac{x}{2}$$ 3. Move all terms involving $$x$$ to one side: $$-\frac{2x}{3} + \frac{x}{2} > 4 - 1$$ $$-\frac{2x}{3} + \frac{x}{2} > 3$$ 4. Find a common denominator (LCM of 3 and 2 is 6): $$-\frac{4x}{6} + \frac{3x}{6} > 3$$ $$-\frac{x}{6} > 3$$ 5. Multiply through by $$-6$$ (reverse the inequality sign): $$x < -18$$ #### Step 1.2: Solve the second inequality $$3 - \frac{x}{4} < x$$ 1. Move $$x$$ to one side: $$3 < x + \frac{x}{4}$$ 2. Combine like terms: $$3 < \frac{4x}{4} + \frac{x}{4}$$ $$3 < \frac{5x}{4}$$ 3. Multiply through by 4: $$12 < 5x$$ 4. Solve for $$x$$: $$x > \frac{12}{5}$$ #### Solution to the first system: The solution is the intersection of the two inequalities: $$x < -18 \quad \text{and} \quad x > \frac{12}{5}.$$ Since these intervals do not overlap, there is **no solution** to this system. --- ### 2. Second system of inequalities: $$\begin{cases} x^2 - 5x + 4 \leq 0 \\ 9 - 4x < 0 \end{cases}$$ #### Step 2.1: Solve the first inequality $$x^2 - 5x + 4 \leq 0$$ 1. Factorize the quadratic: $$x^2 - 5x + 4 = (x - 1)(x - 4)$$ 2. Solve the inequality: $$(x - 1)(x - 4) \leq 0$$ Use a sign chart: - Roots are $$x = 1$$ and $$x = 4$$. - The parabola opens upwards, so the solution is: $$1 \leq x \leq 4$$ #### Step 2.2: Solve the second inequality $$9 - 4x < 0$$ 1. Simplify: $$-4x < -9$$ 2. Divide through by $$-4$$ (reverse the inequality sign): $$x > \frac{9}{4}$$ #### Solution to the second system: The solution is the intersection of the two inequalities: $$1 \leq x \leq 4 \quad \text{and} \quad x > \frac{9}{4}.$$ This gives: $$\frac{9}{4} \leq x \leq 4.$$ --- ### Final Answer: 1. First system: **No solution**. 2. Second system: $$\frac{9}{4} \leq x \leq 4$$. --- Do you need further clarifications or step-by-step explanations? Here are 5 related questions to explore further: 1. How do you solve systems of linear inequalities graphically? 2. What are the methods to solve quadratic inequalities? 3. How can sign charts help in solving polynomial inequalities? 4. Can systems of inequalities have infinite solutions? 5. How do you interpret the solution of inequalities in real-world problems? **Tip:** Always check the feasibility of a solution by testing boundary points in the inequalities.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Inequalities
Linear Equations
Quadratic Equations
Systems of Inequalities
Formulas
Simplification of linear expressions
Finding the roots of a quadratic equation: ax^2 + bx + c = 0
Intersection of inequality solutions
Theorems
Rules of inequalities (multiplying/dividing by negative reverses inequality sign)
Factoring quadratic equations
Suitable Grade Level
Grades 8-10