Solve $5 w+8 \leq 3 w+14$

To solve the inequality 5 w + 8 ≤ 3 w + 14 , we first isolate w through a series of steps, ultimately finding that w ≤ 3 . This means any value less than or equal to 3 will satisfy this inequality. Understanding such inequalities aids in real-world problem-solving.
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Answered by Anonymous | 2025-07-04

Subtract 3 w from both sides: 2 w + 8 ≤ 14 .
Subtract 8 from both sides: 2 w ≤ 6 .
Divide by 2: w ≤ 3 .
The solution is w ≤ 3 .

Explanation

Understanding the Problem We are given the inequality 5 w + 8 ≤ 3 w + 14 and we want to solve for w . This means we want to isolate w on one side of the inequality to find the values of w that satisfy the inequality.

Subtracting 3 w from both sides First, let's subtract 3 w from both sides of the inequality to get the terms with w on one side: 5 w + 8 − 3 w ≤ 3 w + 14 − 3 w

Simplifying the inequality Simplifying both sides, we have: 2 w + 8 ≤ 14

Subtracting 8 from both sides Next, we subtract 8 from both sides to isolate the term with w :
2 w + 8 − 8 ≤ 14 − 8

Simplifying the inequality Simplifying again, we get: 2 w ≤ 6

Dividing both sides by 2 Finally, we divide both sides by 2 to solve for w :
2 2 w ​ ≤ 2 6 ​

The solution This simplifies to: w ≤ 3


Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might have a constraint like 'spending must be less than or equal to your income.' Solving inequalities helps determine how much you can spend while staying within your budget. Similarly, in engineering, inequalities are used to ensure that structures can withstand certain loads or stresses without failing. Inequalities also play a vital role in optimization problems, where you want to maximize profits or minimize costs subject to certain constraints.

Answered by GinnyAnswer | 2025-07-04