Math Problem Statement

Solve the given system of inequalities: - b) {14 - x ≥ 1, x - 10 ≥ 0} - c) {7x + 1 ≥ 0, 2x - 1 > 0} - d) {2(4x + 3) > -2, 4(x - 5) < 16}

Solution

Let's solve the inequalities step by step for b), c), and d).

b)

The system of inequalities is:

14 - x \geq 1 \\ x - 10 \geq 0 \end{cases}$$ **1st Inequality:** $$14 - x \geq 1$$ $$14 - 1 \geq x \implies x \leq 13$$ **2nd Inequality:** $$x - 10 \geq 0$$ $$x \geq 10$$ **Combining the Results:** The solution is where both inequalities are satisfied: $$10 \leq x \leq 13$$ --- ### **c)** The system of inequalities is: $$\begin{cases} 7x + 1 \geq 0 \\ 2x - 1 > 0 \end{cases}$$ **1st Inequality:** $$7x + 1 \geq 0$$ $$7x \geq -1 \implies x \geq -\frac{1}{7}$$ **2nd Inequality:** $$2x - 1 > 0$$ $$2x > 1 \implies x > \frac{1}{2}$$ **Combining the Results:** The stricter condition is $$x > \frac{1}{2}$$ because it overrides $$x \geq -\frac{1}{7}$$. Thus, the solution is: $$x > \frac{1}{2}$$ --- ### **d)** The system of inequalities is: $$\begin{cases} 2(4x + 3) > -2 \\ 4(x - 5) < 16 \end{cases}$$ **1st Inequality:** $$2(4x + 3) > -2$$ Distribute $$2$$: $$8x + 6 > -2$$ Subtract $$6$$: $$8x > -8$$ Divide by $$8$$: $$x > -1$$ **2nd Inequality:** $$4(x - 5) < 16$$ Distribute $$4$$: $$4x - 20 < 16$$ Add $$20$$: $$4x < 36$$ Divide by $$4$$: $$x < 9$$ **Combining the Results:** The solution is where both inequalities are satisfied: $$-1 < x < 9$$ --- ### **Final Solutions:** - **b)** $$10 \leq x \leq 13$$ - **c)** $$x > \frac{1}{2}$$ - **d)** $$-1 < x < 9$$ --- Do you want a detailed explanation of any step? Let me know! Here are 5 related questions to deepen your understanding: 1. How do you graph the solution $$10 \leq x \leq 13$$ on a number line? 2. What happens if one inequality in a system has no solution? 3. How do you solve compound inequalities involving "and" or "or" conditions? 4. How would the solution change if $$x - 10 > 0$$ instead of $$x - 10 \geq 0$$? 5. Can you express $$x > \frac{1}{2}$$ in interval notation? **Tip:** Always simplify and isolate the variable step by step when solving inequalities!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Solving Linear Inequalities
Compound Inequalities
Combining Solutions

Formulas

Basic inequality manipulations: ax + b ≥ c
Distributive property: a(b + c) = ab + ac

Theorems

Properties of inequalities (addition, subtraction, multiplication, division)
Combining ranges of solutions

Suitable Grade Level

Grades 8-10

Related Recommendation

Solving Simultaneous Linear Inequalities: Step-by-Step Guide

Solve a System of Inequalities and Find the Smallest Integer Solution

System of Inequalities: Solving for Feasible Points

Solve the System of Inequalities 4(x-3)-3(x-2)<5 and -6x+7≤0

System of Linear Inequalities: 5x - 4y < 20, 2x + 7y ≥ 28, and 3x + 6y - 6 ≥ y + 21