Math Problem Statement

|10-4y| ≥ 4

Solution

We are given the inequality:

104y4|10 - 4y| \geq 4

Step 1: Understand the absolute value inequality

For absolute value inequalities of the form AB|A| \geq B, where B0B \geq 0, we split it into two cases:

  1. ABA \geq B
  2. ABA \leq -B

So, applying this to our equation:

104y4or104y410 - 4y \geq 4 \quad \text{or} \quad 10 - 4y \leq -4

Step 2: Solve each inequality separately

Case 1: 104y410 - 4y \geq 4

104y410 - 4y \geq 4 Subtract 10 from both sides: 4y410-4y \geq 4 - 10 4y6-4y \geq -6 Now, divide by 4-4 (remember to reverse the inequality when dividing by a negative number): y64=32y \leq \frac{-6}{-4} = \frac{3}{2} So, y32y \leq \frac{3}{2}.

Case 2: 104y410 - 4y \leq -4

104y410 - 4y \leq -4 Subtract 10 from both sides: 4y410-4y \leq -4 - 10 4y14-4y \leq -14 Now, divide by 4-4 (again, reverse the inequality): y144=72y \geq \frac{-14}{-4} = \frac{7}{2} So, y72y \geq \frac{7}{2}.

Step 3: Combine the results

From Case 1, we have y32y \leq \frac{3}{2}, and from Case 2, we have y72y \geq \frac{7}{2}.

Thus, there is no overlap between these two conditions, so the solution to the inequality is:

y32ory72y \leq \frac{3}{2} \quad \text{or} \quad y \geq \frac{7}{2}

Final Solution:

y(,32][72,)y \in (-\infty, \frac{3}{2}] \cup [\frac{7}{2}, \infty)

Would you like further details on any step?


Here are 5 related questions:

  1. How do you solve absolute value equations instead of inequalities?
  2. What happens when the absolute value inequality has a strict inequality, like A<B|A| < B?
  3. Can you explain how to graph absolute value inequalities on a number line?
  4. What is the difference between "and" & "or" conditions in absolute value inequalities?
  5. How would this solution change if the inequality was 104y4|10 - 4y| \leq 4?

Tip: Always remember to flip the inequality when dividing or multiplying by a negative number!

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Math Problem Analysis

Mathematical Concepts

Absolute Value Inequalities
Inequalities
Linear Equations

Formulas

For |A| ≥ B, where B ≥ 0: A ≥ B or A ≤ -B

Theorems

Absolute Value Theorem

Suitable Grade Level

Grades 8-10