rac2x - 1x + 3 > 1 - Coaching Toolbox

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

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

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:

📰 e-7 wedgetail 📰 define sublimation 📰 quest diagnostics payment 📰 Passenger Van Rental Dallas 5585029 📰 Focuses On Immediate Gains And Promises Powerful Advantagesstrong For Gaming Audiences 6379709 📰 Courier Press Evansville Indiana Obituaries 4396005 📰 Best Games Ps5 2025 2294553 📰 Stimulus 2025 7580901 📰 Huntington Park Apartments 3151313 📰 Microsoft Keynote Shocked The World What They Revealed Will Change Everything 2603360 📰 Parking Decks Near Me 3252595 📰 The Ultimate Bloons Tower Defense 3 Guide Thats Winning The Game For Days 2234557 📰 Basketball Philadelphia Schedule 9793051 📰 Playcrazygames 6276056 📰 Twd Characters 9845287 📰 Window 11 Usb 3249766 📰 6 Flags Marine World California 1777290 📰 Learn The Ultimate Trick To Enter Cells Like A Pro In Microsoft Excel 3650543

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