Solve the inequality algebraically.
[tex]2\left(x^2+4\right)\ \textgreater \ 3(x-1)^2-x^2[/tex]
The solution set is $\square$
(Type your answer in interval notation. Simplify your answer. Use integers or fractions for any numbers in the expression.)
Answer (1)
Expand both sides of the inequality: 3(x-1)^2 - x^2"> 2 ( x 2 + 4 ) > 3 ( x − 1 ) 2 − x 2 becomes 2x^2 - 6x + 3"> 2 x 2 + 8 > 2 x 2 − 6 x + 3 .
Simplify the inequality: Subtract 2 x 2 from both sides to get -6x + 3"> 8 > − 6 x + 3 .
Isolate the x term: Subtract 3 from both sides to get -6x"> 5 > − 6 x .
Solve for x: Divide by -6 and flip the inequality sign to get -\frac{5}{6}"> x > − 6 5 , so the solution set is ( − 6 5 , ∞ ) .
Explanation
Analyze the problem We are given the inequality 3(x-1)^2-x^2"> 2 ( x 2 + 4 ) > 3 ( x − 1 ) 2 − x 2 . Our goal is to solve for x and express the solution in interval notation. First, we need to expand both sides of the inequality.
Expand both sides Expanding the left side, we have 2 ( x 2 + 4 ) = 2 x 2 + 8 . Expanding the right side, we have 3 ( x − 1 ) 2 − x 2 = 3 ( x 2 − 2 x + 1 ) − x 2 = 3 x 2 − 6 x + 3 − x 2 = 2 x 2 − 6 x + 3 .
Rewrite the inequality Now we can rewrite the inequality as 2x^2 - 6x + 3"> 2 x 2 + 8 > 2 x 2 − 6 x + 3 .
Simplify the inequality Next, we simplify the inequality by subtracting 2 x 2 from both sides: 2x^2 - 6x + 3 - 2x^2"> 2 x 2 + 8 − 2 x 2 > 2 x 2 − 6 x + 3 − 2 x 2 , which simplifies to -6x + 3"> 8 > − 6 x + 3 .
Isolate the x term Now, we isolate the term with x by subtracting 3 from both sides: -6x + 3 - 3"> 8 − 3 > − 6 x + 3 − 3 , which simplifies to -6x"> 5 > − 6 x .
Solve for x To solve for x , we divide both sides by -6. Remember that when we divide by a negative number, we must flip the inequality sign: − 6 5 < x , which simplifies to -\frac{5}{6}"> x > − 6 5 .
Express the solution in interval notation Finally, we express the solution set in interval notation. Since x is greater than − 6 5 , the solution set is ( − 6 5 , ∞ ) .
Examples
Imagine you're adjusting the temperature in a room. The inequality helps determine the range of temperature settings that keep the room warmer than a certain threshold, ensuring comfort. Similarly, in finance, it can help find investment returns that exceed a minimum target, ensuring profitability. This algebraic approach is useful in scenarios where maintaining a certain condition or exceeding a specific value is crucial.
