rac2x - 1x + 3 > 1

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

🔗 Related Articles You Might Like:

📰 "Breaking: Janice Griffith $10M Lawsuit Thrills Legal Fans and Celebrities! 📰 "The Trial You Can’t Ignore: Janice Griffith Files Shock Lawsuit That Shocked the Public! 📰 "Janice Griffith’s Latest Lawsuit Revealed – Is This the Start of a Major Legal Battle? 📰 Sharming Moment Caught With Someone He Promised Forever 📰 Shattering Expectations The Black Swan Yoga Experience No One Saw Coming 📰 Shazam Cast Revealedyou Wont Believe Who Got The Role 📰 Shazams Casting Shock Youll Find This Choice Unthinkable 📰 She Crushed My Confidencethis Cougar Web Obsession Was Unstoppable 📰 She Didnt Train Athletesshe Rewired Minds And The Truth Was Terrifying 📰 She Raised Him This Way And Now The Evidence Speaks Volumes 📰 She Said No To Beetvand Now Everything She Touches Comes Alive 📰 She Saw Him Oncea Lifetime Lost In A Single Look 📰 She Was Just A Kidnow Barbies Royal Education Changed Everything Try It 📰 She Wasnt Just Cleaning She Was Controlling 📰 She Wiped More Than Floorsshe Silenced Everything 📰 She Wont Watch You Forget Her Name On Your Birthdaysay These Words Aloud 📰 Shes Been Waiting All Yearheres The Warmest Birthday Gift She Deserves 📰 Shoblings From Belowthe Silenced Sounds That Haunt The Underground

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