Why -1 Enters the Domain in Solving Inequalities and Equations with Square Roots

Introduction

In the context of solving complex equations involving square roots, particularly those with multiple inequalities, determining the domain or the set of all possible values for the variable can be intricate. This article explores the steps required to find the domain of a function given by the following system of inequalities and an equation involving square roots:

(begin{cases} 4x^2-9x-5 ge 0 2x^2-x-1 ge 0 x^2-1 ge 0 end{cases})

Step-by-Step Solution

1. Solving the Inequalities Individually

Let's start by solving each inequality separately.

1.1 (4x^2 - 9x - 5 ge 0)

Factoring this, we get:

(4x^2 - 9x - 5 4(x - 5)(x frac{1}{4}))

The solutions to the equality are (x 5) and (x -frac{1}{4}). The inequality is satisfied for:

((-infty, -frac{1}{4}] cup [5, infty))

1.2 (2x^2 - x - 1 ge 0)

Factoring this, we get:

(2x^2 - x - 1 (2x - 1)(x 1))

The solutions to the equality are (x frac{1}{2}) and (x -1). The inequality is satisfied for:

((-infty, -1] cup [frac{1}{2}, infty))

1.3 (x^2 - 1 ge 0)

This factorizes to:

(x^2 - 1 (x - 1)(x 1))

The solutions to the equality are (x 1) and (x -1). The inequality is satisfied for:

((-infty, -1] cup [1, infty))

2. Finding the Intersection of These Solution Sets

Now, let's intersect these solution sets:

[ left((-infty, -frac{1}{4}] cup [5, infty) right) cap left((-infty, -1] cup [frac{1}{2}, infty) right) cap left((-infty, -1] cup [1, infty) right)]

First, we find the intersections of the pairs:

[ left((-infty, -frac{1}{4}] cap (-infty, -1] right) cup left((-infty, -frac{1}{4}] cap [1, infty) right) cup left([-1, infty) cap (-infty, -1] right) cup left([-1, infty) cap [frac{1}{2}, infty) right)]

Which simplifies to:

[ (-infty, -frac{1}{4}] cup emptyset cup emptyset cup [-1, infty)]

Merging these, we get:

((-infty, -frac{1}{4}] cup {-1} cup [1, infty))

3. Verifying -1 as a Solution

At (x -1), the arguments of the square roots are zero, making -1 indeed a valid solution.

4. Solving the Equation

Let's solve the equation:

[sqrt{4x^2 - 9x - 5} sqrt{2x^2 - x - 1}sqrt{x^2 - 1}]

Squaring both sides, we obtain:

[4x^2 - 9x - 5 2x^2 - x - 1(x^2 - 1)sqrt{2x - 1(x - 1)^2}]

Further simplification yields:

[x^2 - 8x 7 2(x - 1)(x 1)sqrt{2x - 1})

Let's simplify the left-hand side:

[(x - 7)(x - 1) 2(x - 1)(x 1)sqrt{2x - 1})

This simplifies to:

[(x - 7)(x - 1) 2(x - 1)(x 1)sqrt{2x - 1})

Removing common factors (x - 1) (noting that (x eq 1) since (x 1) does not satisfy the original inequalities), we get:

[x^2 - 8x 7 2(x 1)sqrt{2x - 1})

5. Finding the Roots

Solving the quadratic equation:

[x^2 - 8x 7 0]

We find the roots using the quadratic formula:

[x frac{8 pm sqrt{64 - 28}}{2} frac{8 pm sqrt{36}}{2} frac{8 pm 6}{2})

This gives us:

[x 7, -frac{1}{7})

Only (x 5) satisfies all conditions, hence the solutions are (-1) and (5).

6. Verifying the Domain

For the first inequality, we complete the square:

[4x^2 - 9x - 5 4left(x - frac{9}{8}right)^2 - frac{1}{16} ge 0]

The argument is nonnegative if:

[x in (-infty, -frac{5}{4}] cup [-1, infty)]

For the second inequality:

[2x^2 - x - 1 (2x - 1)(x 1) ge 0]

The argument is nonnegative if:

[x in (-infty, -1] cup [frac{1}{2}, infty)]

For the third inequality:

[x^2 - 1 (x - 1)(x 1) ge 0]

The argument is nonnegative if:

[x in (-infty, -1] cup [1, infty)]

Merging these, we find the intersection:

[x in (-infty, -frac{5}{4}] cup {-1} cup [1, infty)]

Conclusion

The use of square roots in mathematical expressions often introduces specific conditions that must be considered, such as ensuring that the arguments of the square roots are nonnegative. In this case, -1 enters the domain because it is a solution to the original equations and satisfies the required inequalities. This example highlights the importance of checking each step and considering all possible solutions rigorously to ensure the final domain is accurate and complete.