Solve this inequality: $8 z+3-2 z<51$
A) $z<8$
B) $z<14$
C) $z<9$
D) $z<5.4$
Answer (1)
Combine like terms: 8 z − 2 z + 3 < 51 becomes 6 z + 3 < 51 .
Subtract 3 from both sides: 6 z < 48 .
Divide both sides by 6: z < 8 .
The solution to the inequality is z < 8 .
Explanation
Understanding the Inequality We are given the inequality 8 z + 3 − 2 z < 51 . Our goal is to isolate z on one side of the inequality to find the solution set.
Combining Like Terms First, we combine like terms on the left side of the inequality. We have 8 z − 2 z = 6 z , so the inequality becomes 6 z + 3 < 51 .
Subtracting the Constant Next, we subtract 3 from both sides of the inequality to isolate the term with z . This gives us 6 z + 3 − 3 < 51 − 3 , which simplifies to 6 z < 48 .
Dividing to Solve for z Now, we divide both sides of the inequality by 6 to solve for z . This gives us 6 6 z < 6 48 , which simplifies to z < 8 .
Final Answer Therefore, the solution to the inequality is z < 8 . Comparing this to the given options, we see that option A is the correct answer.
Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting and resource allocation. For example, if you have a limited budget for a project, you can use inequalities to determine the maximum amount you can spend on different aspects of the project while staying within your budget. Similarly, in manufacturing, inequalities can help determine the range of acceptable values for product dimensions to ensure quality control.
