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.

Solved integral x + 1 / x^3 - 2x^2 + x dx x (x^2 - 2x +1) x | Chegg.com

Image Gallery

Solved integral x + 1 / x^3 - 2x^2 + x dx x (x^2 - 2x +1) x | Chegg.com
Which equation has the same solutions as 2x^2 + x -3=0. Is it's: (2x-1 ...
Solved Sketch r(x)=-2x3(1-x)(x+3)2 using zeros, | Chegg.com
Solved 2x -1/x + 3 - x -2/2x + 1 | Chegg.com
Solved solution of (2x−1)(x+3)

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

Behind the Glow: Shree Ram Krishna Sweeps with Diamonds for the Global Market!
India’s Diamond Power: The Shree Ram Krishna Story Behind the JLF Dual Award!
From Punjab to the World: How Shree Ram Krishna’s Exports Shook the Jewelry Industry!

🔗 Related Articles You Might Like:

📰 How to Add a Watermark in Word 📰 How to Add a Youtube Video to Powerpoint 📰 How to Add Accent Marks in Word 📰 The End Of The Watch Movie Left Fans Screamingheres The Mind Blowing Finale Reveal 9690634 📰 Film Ninjago 50394 📰 Roblox Studio Toolbox 3390338 📰 Fragrant 1385064 📰 Video Game Fear 2 5653653 📰 Woodstock Pizza Santa Cruz 7440543 📰 Top 10 Free Xmas Games You Can Play Instantlyno Cost All Joy 1129678 📰 The Shocking Truth What Nemesis Prime Is Waiting To Destroy Your World 8020006 📰 The Cartoon Girl Who Disappeared Without A Trace You Wont Believe Her Secret 4851154 📰 Epl Man Utd Vs Newcastle 6397171 📰 Is This The Biggest Nude Celebrity Photo Scandal Yet Dont Miss This Reveal 9359039 📰 Waiting Has Ended Half Life 2 Re Release Date Revealed In 2025 8918887 📰 Verizon Home Wifi Extender 507349 📰 The Desperate Choice Fueled By Fire On Both Sides 8024443 📰 1964 Worlds Fair 7278045

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