Understanding the Inequality: 3x + 7 < 16 – A Step-by-Step Guide

Solving mathematical inequalities is a fundamental skill in algebra, essential for students, educators, and anyone looking to strengthen their problem-solving abilities. One common type of problem students encounter is linear inequalities like 3x + 7 < 16. This article breaks down how to solve this inequality step-by-step, explains its real-world applications, and provides tips to master similar problems.

Understanding the Context

What Does “3x + 7 < 16” Mean?

The inequality 3x + 7 < 16 tells us that when you multiply an unknown value x by 3, add 7, the result is less than 16. Our goal is to isolate x to find the range of values that satisfy the condition.

Step-by-Step Solution

3x + 7 < 16

Key Insights

Let’s solve 3x + 7 < 16 systematically:

Step 1: Subtract 7 from both sides.
This eliminates the constant on the left side:
$$
3x + 7 - 7 < 16 - 7
$$
$$
3x < 9
$$

Step 2: Divide both sides by 3.
Since 3 is positive, the inequality direction stays the same:
$$
rac{3x}{3} < rac{9}{3}
$$
$$
x < 3
$$

The Solution

Continue Reading

A train travels 180 miles at 60 mph, then continues for another 120 miles at 80 mph. What is the train's average speed for the entire journey?
Time for the first segment: \(180 \div 60 = 3\) hours.
Time for the second segment: \(120 \div 80 = 1.5\) hours.

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Final Thoughts

The inequality 3x + 7 < 16 is true for all real numbers x less than 3. In interval notation, the solution is:
x ∈ (-∞, 3)

Real-World Applications

Understanding inequalities like 3x + 7 < 16 helps in everyday decision-making and problem solving. For example:

  • Budgeting: If a product costs $3 each plus a $7 service fee, and you want to stay under $16, solving 3x + 7 < 16 tells you how many items (x) you can buy.
  • Science: When measuring temperature ranges or chemical concentrations that cannot exceed a threshold.
  • Goal Setting: Determining how many hours (x) you can study under time or rest constraints.

Tips to Master Linear Inequalities

  1. Always perform the same operation on both sides to maintain equality or inequality.
  2. Pay attention to the direction of the inequality: multiplying or dividing by a negative reverses the sign; dividing by a positive only.
  3. Practice with word problems to translate real scenarios into mathematical inequalities.
  4. Use number lines to visualize solutions like x < 3.
  5. Check your solution by substituting a number less than 3 back into the original inequality.

Final Thoughts