Solving Inequalities: Techniques and Examples

When faced with the challenge of solving inequalities, it is crucial to understand the underlying techniques and the specific conditions that must be met.

Identifying the Numerator and Denominator

The process of solving an inequality often starts by understanding the components of the expression. Consider the expression:

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

The numerator can be written as:

(x^2 - 9 (x - 3)(x 3) )

Thus, the inequality we need to solve is:

(frac{(x - 3)(x 3)}{x - 1} geq 0 )

Case Analysis

We need to analyze the expression by considering different cases based on the critical points where the expression changes sign.

Case I: When (x > 3)

For this case, both x - 3 and x 3 are positive. Therefore:

(x - 3) > 0 (x 3) > 0 x - 1 > 0

This means the expression is positive when (x > 3).

Case II: When (x

For this case, both x - 3 and x 3 are negative, and x - 1 is also negative. Therefore:

(x - 3) (x 3) x - 1

This means the expression is positive when (x

Case III: When (-3

In this case, x - 3 is negative, x 3 is positive, and x - 1 is negative. Therefore:

(x - 3) (x 3) > 0 x - 1

This results in a negative expression.

Case IV: When (-1

Here, x - 3 is negative, x 3 is positive, and x - 1 is negative. Therefore:

(x - 3) (x 3) > 0 x - 1

This also results in a negative expression.

Case V: When (1

In this case, x - 3 is negative, x 3 is positive, and x - 1 is positive. Therefore:

(x - 3) (x 3) > 0 x - 1 > 0

This means the expression is negative.

Combining all the cases:

The solution is:

x leq -3 x > 1 x eq 1

General Guidelines for Solving Inequalities

Respect the domain of the expression: Always check for any values that make the denominator zero (e.g., (x eq 1) in this case). Factorize the numerator to simplify the expression and identify critical points. Test intervals between critical points to determine where the expression is positive or negative. Combine all intervals where the expression meets the inequality condition.

Example: Further Solving with Zero and Denominator Constraints

Consider the expression:

(frac{x^2 - 4}{x - 2} )

First, factorize the numerator:

(frac{(x - 2)(x 2)}{x - 2} )

For (x eq 2), the expression simplifies to:

(x 2 )

Solve the inequality:

(x 2 geq 0 )

This gives:

(x geq -2 )

Remember to exclude (x 2) since it makes the denominator zero. Therefore, the final solution is:

(x geq -2, x eq 2 )

Keywords: inequality, algebraic solution, factorization