- Expand the inequality: \(30x + 20 > 14x - 28\).
- Move variables to one side and constants to the other: \(16x > -48\).
- Divide by the coefficient: \(x > -3\).
- The correct option is \(\boxed{c}\).
Explanation
- Analysis of the Problem
We are given the inequality \(10(3x + 2) > 7(2x - 4)\). The objective is to solve for \(x\) and identify which of the provided options—\(x > -4\), \(x < -4\), \(x > -3\), or \(x < -3\)—correctly represents the solution set. To do this, we will use algebraic manipulation to isolate \(x\) on one side of the inequality.
- Expanding and Grouping Variables
First, we expand both sides of the inequality using the distributive property:
\[10 \cdot 3x + 10 \cdot 2 > 7 \cdot 2x + 7 \cdot (-4)\]
\[30x + 20 > 14x - 28\]
Now, we collect the terms involving \(x\) on one side by subtracting \(14x\) from both sides:
\[30x - 14x + 20 > -28\]
\[16x + 20 > -28\]
- Solving for x
Next, we isolate the \(x\) term by subtracting 20 from both sides:
\[16x > -28 - 20\]
\[16x > -48\]
Finally, we solve for \(x\) by dividing both sides by 16:
\[x > \frac{-48}{16}\]
\[x > -3\]
This matches option (c).
Examples
Understanding how to solve linear inequalities is essential for many real-world scenarios, such as budgeting or resource allocation. For example, if a store offers two different promotional deals, you can use inequalities to determine which range of spending qualifies you for a better discount. If one deal requires a base fee plus a rate based on the number of items, and you want to know when that deal becomes more cost-effective than another, you can set up and solve an inequality just like this one to make an informed financial decision.