$4(3-0.1 x)+\frac{1}{5}(4 x-25) \leq \frac{2}{3} x+\frac{3}{5}-(15+0.6 x)$
Answer (2)
To solve the given inequality, first expand and simplify both sides, then isolate the x terms on one side. After some algebra, the solution is x ≤ − 5 321 or x ≤ − 64.2 .
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Expand both sides of the inequality and simplify.
Combine like terms to isolate x terms on one side and constants on the other.
Solve for x, remembering to flip the inequality sign when multiplying or dividing by a negative number.
The solution to the inequality is x ≤ − 5 321 .
Explanation
Understanding the Problem We are given the inequality 4 ( 3 − 0.1 x ) + 5 1 ( 4 x − 25 ) ≤ 3 2 x + 5 3 − ( 15 + 0.6 x ) Our goal is to solve this inequality for x .
Expanding Both Sides First, we expand both sides of the inequality: 12 − 0.4 x + 5 4 x − 5 ≤ 3 2 x + 5 3 − 15 − 0.6 x 7 − 0.4 x + 0.8 x ≤ 3 2 x − 0.6 x + 5 3 − 15 7 + 0.4 x ≤ 3 2 x − 5 3 x − 5 72 7 + 0.4x \leq \(\frac{10-9}{15}\)x - \frac{72}{5} 7 + 0.4 x ≤ 15 1 x − 5 72
Isolating x Terms Next, we want to isolate the x terms on one side and the constant terms on the other side. To do this, we subtract 0.4 x from both sides and add 5 72 to both sides: 7 + 5 72 ≤ 15 1 x − 0.4 x 5 35 + 72 ≤ 15 1 x − 5 2 x 5 107 ≤ 15 1 x − 15 6 x 5 107 ≤ − 15 5 x 5 107 ≤ − 3 1 x
Solving for x Now, we solve for x by multiplying both sides by − 3 . Remember to flip the inequality sign since we are multiplying by a negative number: − 3 × 5 107 ≥ x − 5 321 ≥ x x ≤ − 5 321 x ≤ − 64.2
Final Answer Therefore, the solution to the inequality is x ≤ − 5 321 or x ≤ − 64.2 .
Examples
Linear inequalities are used in various real-world scenarios, such as determining budget constraints. For example, if you have a certain amount of money to spend on groceries, and each item has a different price, you can use a linear inequality to determine how many of each item you can buy without exceeding your budget. Similarly, in business, linear inequalities can help determine the optimal production levels to maximize profit while staying within resource limitations. Understanding and solving linear inequalities is a fundamental skill with broad applications in economics, finance, and everyday decision-making.
