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: $\qquad$
$x \leq-9$
$x \geq-9$
$x \leq-\frac{1}{9}$
$x \geq-\frac{1}{9}$
Answer (1)
Distribute and combine like terms to simplify the inequality.
Isolate the variable term on one side of the inequality.
Divide both sides by the coefficient of the variable, remembering to flip the inequality sign if dividing by a negative number.
The solution to the inequality is x ≥ − 9 .
Explanation
Analyzing the Problem We are given the inequality 4 ( x − 3 ) + 4 ≤ 10 + 6 x and the following steps:
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
The division property of inequality: x ≤ − 9 , xg e − 9 , x ≤ − 9 1 , xg e − 9 1
We need to find the correct missing step between step 4 and step 5.
Solving the Inequality To find the missing step, we need to solve the inequality − 2 x ≤ 18 . We can do this by dividing both sides of the inequality by − 2 . Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign.
Applying the Division Property Dividing both sides of − 2 x ≤ 18 by − 2 , we get: − 2 − 2 x g e − 2 18 xg e − 9
Finding the Missing Step Therefore, the missing step is the division property of inequality, which gives us xg e − 9 .
Final Answer The missing step is xg e − 9 .
Examples
Understanding inequalities is crucial in various real-life scenarios. For instance, when budgeting, you might want to ensure that your expenses do not exceed your income, which can be represented as an inequality. Similarly, in manufacturing, quality control often involves ensuring that products meet certain specifications within a tolerance range, which can also be expressed using inequalities. This problem demonstrates how to solve a linear inequality, a fundamental skill for making informed decisions in many practical situations.
