Solve the following inequality using the algebraic approach: $5 x-1<2 x+11$
A. $x<\frac{12}{7}$
B. $x<\frac{10}{3}$
C. $x<\frac{10}{7}$
D. $x<4$
Answer (2)
Subtract 2 x from both sides: 3 x − 1 < 11 .
Add 1 to both sides: 3 x < 12 .
Divide both sides by 3 : x < 4 .
The solution to the inequality is x < 4 .
Explanation
Understanding the Inequality We are given the inequality 5 x − 1 < 2 x + 11 . Our goal is to isolate x on one side of the inequality to find the solution.
Subtracting 2x from Both Sides First, let's subtract 2 x from both sides of the inequality: 5 x − 1 − 2 x < 2 x + 11 − 2 x This simplifies to: 3 x − 1 < 11
Adding 1 to Both Sides Next, we add 1 to both sides of the inequality: 3 x − 1 + 1 < 11 + 1 This simplifies to: 3 x < 12
Dividing by 3 Now, we divide both sides of the inequality by 3 :
3 3 x < 3 12 This simplifies to: x < 4
Finding the Solution The solution to the inequality is x < 4 . Comparing this to the given options, we see that option d matches our solution.
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might want to ensure that your expenses are less than your income, which can be represented as an inequality. Similarly, in cooking, you might need to keep the temperature of an oven below a certain threshold to prevent burning the food. Inequalities also play a significant role in optimization problems, such as maximizing profit or minimizing cost, where constraints are often expressed as inequalities.
The solution to the inequality 5 x − 1 < 2 x + 11 is x < 4 . Thus, the correct answer is option D: x < 4 .
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