3x + 7 < 16 - Decision Point
Understanding the Inequality: 3x + 7 < 16 – A Step-by-Step Guide
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
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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
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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
- Always perform the same operation on both sides to maintain equality or inequality.
- Pay attention to the direction of the inequality: multiplying or dividing by a negative reverses the sign; dividing by a positive only.
- Practice with word problems to translate real scenarios into mathematical inequalities.
- Use number lines to visualize solutions like x < 3.
- Check your solution by substituting a number less than 3 back into the original inequality.
