Solve the rational inequality. (1)/(x) ≤(1)/(2 x-1) |...

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one rational function is compared to another, often through inequality signs like ?, ?, <, or >. Solving these requires an understanding of both the numerator and denominator, including where the function is undefined due to division by zero, as well as how the overall sign of the rational expression changes across different intervals.

Domain Restrictions

When working with rational expressions, it is essential to first determine the domain of the function by identifying values that make any denominator zero. The domain restrictions prevent the inclusion of values that would result in undefined expressions, and these exclusions must be considered when interpreting the solution set of the inequality.

Critical Points

Critical points are the values at which either the numerator or the denominator is zero. These points segment the number line into intervals where the rational expression may change sign. Identifying these critical values is a crucial step in solving rational inequalities because they indicate potential boundaries for the solution set and help in constructing a sign chart.

Sign Chart Analysis

Sign chart analysis involves selecting test points in the intervals determined by the critical points to evaluate the overall sign of the rational expression. This method helps in deciding whether the expression satisfies the inequality in each interval, leading to an accurate solution set.

Handling Inequalities Involving Variable Denominators

When solving inequalities with expressions that have denominators containing variables, extra caution is needed since multiplying or dividing both sides by an expression whose sign is not constant can reverse the inequality. A careful approach involves analyzing the sign of the denominator across different intervals or using methods like creating a common denominator, ensuring that the inequality is manipulated correctly without losing or misrepresenting parts of the solution set.

Transcript

00:01 In this question, we want to find the values of x for which 1 over x is less than or equal to 1 over 2x minus 1.

00:13 To start off, make sure to not multiply both sides by the denominators in this question, because when we're dealing with an inequality such as this, we have to switch the sign for negative numbers.

00:26 Or we switch the sign when we multiply or divide by a negative number.

00:30 And since x and 2x minus 1 can be both positive and negative, we don't want to complicate things by multiplying by them.

00:42 If we do, we would create many cases in which we would have to either switch or keep the sign or the direction of our greater than or less than sign.

01:03 Instead, let's try to figure this out in a different way.

01:09 First of all, why do we want to multiply by the denominators? well, when we multiply by the denominators, we get just two polynomials on each side, or one polynomial on each side.

01:23 We get two polynomials in the end, and that helps us get zero on one side and just a single polynomial on the other.

01:35 Now, since we can't multiply by the denominators, we can do the next best thing.

01:42 Get zero on the right side right away.

01:47 That is, we subtract whatever is on the right side from both sides and get a rational expression on the left with and zero on the right.

01:58 Now, since we can't just get a polynomial expression on the left side, we can get a rational expression on the left side and still are zero on the right side.

02:09 So that we can analyze where this rational expression is less than or equal to zero...

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