Solve this inequality: $8 z+3-2 z<51$
A) $z<9$
B) $z<14$
C) $z<5.4$
D) $z<8$
Answer (1)
Combine like terms: 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 Problem We are given the inequality 8 z + 3 − 2 z < 51 and need to find the range of values for z that satisfies it. The inequality involves a linear expression in z .
Combining Like Terms First, we combine like terms on the left side of the inequality:
8 z − 2 z + 3 < 51
6 z + 3 < 51
Isolating the Variable Term Next, we subtract 3 from both sides of the inequality to isolate the term with z :
6 z + 3 − 3 < 51 − 3
6 z < 48
Solving for z Now, we divide both sides of the inequality by 6 to solve for z :
6 6 z < 6 48
z < 8
Final Answer Therefore, the solution to the inequality is z < 8 .
Examples
Understanding inequalities is crucial in various real-life situations, such as budgeting. For instance, if you have a fixed amount of money to spend on groceries and you know the price of each item, you can use inequalities to determine how many of each item you can buy without exceeding your budget. Similarly, in business, inequalities can help determine the optimal production levels to maximize profit while staying within resource constraints.
