Solve the inequality.
[tex]
\begin{array}{c}
2(4 x-3) \geq-3(3 x)+5 x \\
0 x \geq 0.5 \\
x \geq 2 \\
0(-\infty, 0.5) \\
(-\infty, 2)
\end{array}
[/tex]
Answer (1)
Expand the inequality: = " -3(3x) + 5x"> 2 ( 4 x − 3 ) " >= " − 3 ( 3 x ) + 5 x becomes = " -9x + 5x"> 8 x − 6" >= " − 9 x + 5 x .
Simplify the inequality: = " -4x"> 8 x − 6" >= " − 4 x .
Isolate x: = " 6"> 12 x " >= "6 .
Solve for x: = " \frac{{1}}{{2}}"> x " >= " 2 1 . The solution set is [ 2 1 , ∞ ) .
Explanation
Analyzing the Problem Let's analyze the given inequality and the provided steps to identify the errors and find the correct solution.
Stating the Inequality The given inequality is = " -3(3x) + 5x"> 2 ( 4 x − 3 ) " >= " − 3 ( 3 x ) + 5 x . The provided steps seem to have errors in the simplification and solution process. Let's solve it correctly.
Expanding the Inequality First, expand both sides of the inequality: = " -3(3x) + 5x"> 2 ( 4 x − 3 ) " >= " − 3 ( 3 x ) + 5 x becomes = " -9x + 5x"> 8 x − 6" >= " − 9 x + 5 x
Simplifying the Right Side Simplify the right side: = " -4x"> 8 x − 6" >= " − 4 x
Adding 4x to Both Sides Add 4 x to both sides: = " -4x + 4x"> 8 x + 4 x − 6" >= " − 4 x + 4 x which simplifies to = " 0"> 12 x − 6" >= "0
Adding 6 to Both Sides Add 6 to both sides: = " 0 + 6"> 12 x − 6 + 6" >= "0 + 6 which simplifies to = " 6"> 12 x " >= "6
Dividing by 12 Divide both sides by 12: =" \frac{{6}}{{12}}"> 12 12 x " >= " 12 6 which simplifies to =" \frac{{1}}{{2}}"> x " >= " 2 1
Expressing the Solution The solution to the inequality is = " \frac{{1}}{{2}}"> x " >= " 2 1 . In interval notation, this is [ 2 1 , ∞ ) .
Final Answer The correct solution is = " \frac{{1}}{{2}}"> x " >= " 2 1 , or in interval notation, [ 2 1 , ∞ ) .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, consider a delivery service that guarantees delivery within a certain time frame. If the delivery time exceeds this limit, customers receive a discount. Solving inequalities helps the company determine the maximum delay they can afford before offering a discount, ensuring they remain profitable while maintaining customer satisfaction. Similarly, in budgeting, inequalities can help determine how much money can be spent on different categories while staying within a financial limit. These applications highlight the practical importance of mastering inequalities.
