What is the missing step in solving the inequality $4(x-3)+4 \leq 10+6 x$?
1. The distributive property: $4 x-12+4 \leq 10+6 x$
2. Combine like terms: $4 x-8 \leq 10+6 x$
3. The addition property of inequality: $4 x \leq 18+6 x$
4. The subtraction property of inequality: $-2 x \leq 18$
5. The division property of inequality:
$x \leq-9$
$x \geq-9$
$x \leq-\frac{1}{9}$
$x \geq \frac{1}{9}$
Answer (2)
Distribute and combine like terms to get 4 x − 8 ≤ 10 + 6 x .
Isolate x terms on one side: − 2 x ≤ 18 .
Divide by − 2 , remembering to flip the inequality sign: x ≥ − 9 .
The solution to the inequality is x ≥ − 9 .
Explanation
Problem Analysis We are given the inequality 4 ( x − 3 ) + 4 ≤ 10 + 6 x and the steps taken to solve it. Our goal is to identify the missing step, specifically the correct application of the division property of inequality.
Reviewing the Steps The given steps are:
Distributive property: 4 x − 12 + 4 ≤ 10 + 6 x
Combine like terms: 4 x − 8 ≤ 10 + 6 x
Addition property of inequality: 4 x ≤ 18 + 6 x
Subtraction property of inequality: − 2 x ≤ 18
Division property of inequality: x ≤ − 9 or x ≥ − 9 or x ≤ − 9 1 or x ≥ 9 1
Applying the Division Property To isolate x , we need to divide both sides of the inequality − 2 x ≤ 18 by − 2 . Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign.
Performing the Division Dividing both sides of − 2 x ≤ 18 by − 2 , we get: − 2 − 2 x ≥ − 2 18 x ≥ − 9
Conclusion Therefore, the missing step is x ≥ − 9 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, when budgeting, you might need to ensure that your expenses are less than or equal to your income. Similarly, in engineering, you might need to ensure that a structure can withstand a certain amount of weight or stress. Inequalities help us define and solve these types of problems, ensuring we stay within safe and practical limits.
The missing step in solving the inequality is dividing both sides of − 2 x ≤ 18 by − 2 and flipping the inequality sign, resulting in x ≥ − 9 . Therefore, the correct choice from the options given is x ≥ − 9 .
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