Select the best answer for the question.
12. Solve the following inequality: $38<4 x+3+7-3 x$.
A. $x>28$
B. $x>4$
C. $x<28$
D. $x<4$
Answer (1)
Combine like terms: 38 < 4 x + 3 + 7 − 3 x becomes 38 < x + 10 .
Isolate x by subtracting 10 from both sides: 38 − 10 < x + 10 − 10 .
Simplify the inequality: 28 < x .
The solution is 28"> x > 28 , so the answer is 28}"> x > 28 .
Explanation
Understanding the Inequality We are given the inequality 38 < 4 x + 3 + 7 − 3 x . Our goal is to solve for x .
Simplifying the Inequality First, we simplify the right side of the inequality by combining like terms. We have 4 x − 3 x = x and 3 + 7 = 10 . So the inequality becomes 38 < x + 10 .
Isolating x Next, we want to isolate x . To do this, we subtract 10 from both sides of the inequality: 38 − 10 < x + 10 − 10 , which simplifies to 28 < x .
Finding the Solution Therefore, the solution to the inequality is 28"> x > 28 . Comparing this to the given options, we see that option A is the correct answer.
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, suppose you are managing a budget and need to ensure that your expenses do not exceed your income. If your income is represented by a fixed number and your expenses are a function of a variable (e.g., the number of items you purchase), you can set up an inequality to determine the maximum value of the variable that keeps your expenses within your income. Similarly, in engineering, inequalities are used to set safety margins and ensure that structures can withstand certain loads or stresses. Inequalities also play a vital role in optimization problems, where the goal is to maximize or minimize a certain quantity subject to constraints.
