Math Problem Statement

Solve the inequality for x: a(bx + c) ≥ bc when a, b, c > 0

Solution

The problem is asking to solve the inequality for xx:

a(bx+c)bca(bx + c) \geq bc where a,b,c>0a, b, c > 0.

Let's break it down:

Step 1: Expand the inequality

Expand the left-hand side: a(bx+c)=abx+aca(bx + c) = abx + ac So the inequality becomes: abx+acbcabx + ac \geq bc

Step 2: Move terms involving xx to one side

Subtract acac from both sides: abxbcacabx \geq bc - ac

Step 3: Factor the right-hand side

The right-hand side can be factored as: abxc(ba)abx \geq c(b - a)

Step 4: Solve for xx

Now, divide both sides by abab (since a>0a > 0 and b>0b > 0, we know ab>0ab > 0, so the direction of the inequality remains the same): xc(ba)abx \geq \frac{c(b - a)}{ab}

Step 5: Check answer options

From the options given in the image, the correct answer corresponds to: xc(ba)abx \geq \frac{c(b - a)}{ab} which matches the first option.

Final Answer:

The correct option is the first one: xc(ba)abx \geq \frac{c(b - a)}{ab}


Do you have any questions or would you like more details on any step?

Here are 5 related questions to deepen your understanding:

  1. How would the inequality change if aa or bb were negative?
  2. What happens to the solution if a=ba = b?
  3. How does solving inequalities differ from solving equalities?
  4. Can you generalize this method for other inequalities involving linear expressions?
  5. How would the steps differ if the inequality symbol was \leq instead of \geq?

Tip: When solving inequalities, always pay attention to the direction of the inequality, especially when multiplying or dividing by negative numbers.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Linear Algebra

Formulas

a(bx + c) ≥ bc
abx + ac ≥ bc
x ≥ c(b-a) / ab

Theorems

Basic properties of inequalities
Factoring and expanding linear expressions

Suitable Grade Level

Grades 10-12