rac2x - 1x + 3 > 1 - Abu Waleed Tea
Solving the Inequality rac²(2x – 1)(x + 3) > 1: A Step-by-Step Guide
Solving the Inequality rac²(2x – 1)(x + 3) > 1: A Step-by-Step Guide
When faced with the inequality rac²(2x – 1)(x + 3) > 1, many students and math enthusiasts wonder how to approach it efficiently. This article walks you through solving the inequality rac²(2x – 1)(x + 3) > 1 step-by-step, including key concepts and common pitfalls to avoid.
Understanding the Context
Understanding the Inequality
The inequality to solve is:
rac²(2x – 1)(x + 3) > 1
Here, rac²(2x – 1)(x + 3) means [ (2x – 1)(x + 3) ]², the square of the expression (2x – 1)(x + 3). This quadratic expression is inside a square, making it non-negative regardless of the signs of the factors. The inequality compares this squared expression to 1, so we're essentially solving when a squared term exceeds 1.
Key Insights
Step 1: Rewrite the Inequality Clearly
Start by clearly writing the inequality in standard form:
(2x – 1)(x + 3)² > 1
This step makes it easier to analyze the behavior of the expression.
Step 2: Move All Terms to One Side
Final Thoughts
To prepare solving, bring 1 to the left side:
(2x – 1)(x + 3)² – 1 > 0
Now we want to solve when this expression is greater than zero.
Step 3: Analyze the Function as a Combined Function
Let:
f(x) = (2x – 1)(x + 3)² – 1
Our goal: solve f(x) > 0.
First, expand (x + 3)²:
==> (x + 3)² = x² + 6x + 9
Now substitute:
f(x) = (2x – 1)(x² + 6x + 9) – 1
Multiply out:
f(x) = (2x – 1)(x² + 6x + 9) – 1
= 2x(x² + 6x + 9) – 1(x² + 6x + 9) – 1
= 2x³ + 12x² + 18x – x² – 6x – 9 – 1
= 2x³ + 11x² + 12x – 10
So the inequality becomes:
2x³ + 11x² + 12x – 10 > 0
