The inequality $3 x-11<11 x+18$ is equivalent to:
a. $x>-\frac{29}{8}$
b. $x>\frac{29}{8}$
C. $x<-\frac{29}{8}$
d. $x<\frac{29}{8}$
Answer (1)
Subtract 3 x from both sides: − 11 < 8 x + 18 .
Subtract 18 from both sides: − 29 < 8 x .
Divide both sides by 8: − 8 29 < x .
Rewrite the inequality: -\frac{29}{8}"> x > − 8 29 .
The final answer is -\frac{29}{8}}"> x > − 8 29
Explanation
Understanding the Inequality We are given the inequality 3 x − 11 < 11 x + 18 . Our goal is to isolate x on one side of the inequality to find an equivalent expression.
Subtracting 3 x First, let's subtract 3 x from both sides of the inequality: 3 x − 11 − 3 x < 11 x + 18 − 3 x This simplifies to: − 11 < 8 x + 18
Subtracting 18 Next, we subtract 18 from both sides of the inequality: − 11 − 18 < 8 x + 18 − 18 This simplifies to: − 29 < 8 x
Dividing by 8 Now, we divide both sides of the inequality by 8: 8 − 29 < 8 8 x This simplifies to: − 8 29 < x
Rewriting the Inequality Finally, we can rewrite the inequality as: -\frac{29}{8}"> x > − 8 29
Final Answer Comparing our result with the given options, we see that it matches option a. Therefore, the inequality 3 x − 11 < 11 x + 18 is equivalent to -\frac{29}{8}"> x > − 8 29 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might use inequalities to determine how much you can spend on different items while staying within your income. If you have a limited amount of money, say M , and you want to buy items x and y with prices p x and p y respectively, the inequality p x ⋅ x + p y ⋅ y < M helps you determine the possible quantities of x and y you can afford. Solving and understanding such inequalities allows you to make informed financial decisions.
