Solve the inequality $16-7 a \geq -33$
A) $a \leq 7$
B) $a>-7$
C) $a \geq 7$
D) $a<-7$
Answer (2)
Subtract 16 from both sides: − 7 a ≥ − 49 .
Divide both sides by -7 and flip the inequality sign: a ≤ 7 .
The solution to the inequality is a ≤ 7 .
The final answer is a ≤ 7 .
Explanation
Understanding the Inequality We are given the inequality 16 − 7 a ≥ − 33 . Our goal is to isolate a and solve the inequality.
Isolating the Term with 'a' First, we subtract 16 from both sides of the inequality to isolate the term with a : 16 − 7 a − 16 ≥ − 33 − 16 This simplifies to: − 7 a ≥ − 49
Dividing by -7 (and Flipping the Sign) Next, we divide both sides of the inequality by -7. Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign: − 7 − 7 a ≤ − 7 − 49
Solution This simplifies to: a ≤ 7
Final Answer Therefore, the solution to the inequality 16 − 7 a ≥ − 33 is a ≤ 7 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're managing a budget and need to ensure your expenses don't exceed your income. If your income is represented by a fixed number and your expenses are related to a variable, solving an inequality helps you determine the maximum value of that variable to stay within your budget. Similarly, in manufacturing, inequalities are used to maintain quality control by ensuring that products meet certain specifications within acceptable ranges. This problem demonstrates a fundamental skill in managing constraints and making informed decisions.
The solution to the inequality 16 − 7 a ≥ − 33 is a ≤ 7 . After isolating the term containing 'a' and dividing by -7 while reversing the inequality sign, we arrive at this conclusion. The correct answer is option A, a ≤ 7 .
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