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.

rac{2x - 1}{x + 3} > 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

Continue Reading

Jesus Art Exposed: The Shocking Truth About These Divine Masterpieces That’re Going Viral!
Why This Jesus Art is Suddenly Massively Popular—Watch the Numbers Spike!
The Most Stunning Jesus Art Installations You Need to See Before It’s Too Late!

🔗 Related Articles You Might Like:

📰 Does Dr. Otto Octavius Have the Surprising Cure Proven to Reverse Aging? 📰 Shocking Journey Inside the Mind of Dr. Otto Octavius: The Doctor Who Blows Minds! 📰 Dr. Otto Octavius Exposed the Hidden Truth—Are You Ready for the Discovery? 📰 Trucker Hats For Women 📰 Trucker Womens Hats 📰 True Autumn Color Palette 📰 True Crime Streets Of La 📰 True Detective Season 3 📰 True Detective Third Season 📰 True North Vbs 📰 True Or False Questions 📰 True Scary Films 📰 True Spring Color Palette 📰 True Summer Color Palette 📰 True Winter Color Palette 📰 Trueblood Hbo 📰 Trump Movie 📰 Trumpet Fingering Chart

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