Solve each inequality algebraically. (x^2(3+x)(x+4))/((x+5)(x-1)) ≥0

Name: Problem 45, Chapter 5: College Algebra
Uploaded: 2020-05-21T14:23:47-08:00
Duration: 3 min 40 s
Channel: Ankit Gupta
Description: Solve each inequality algebraically. (x^2(3+x)(x+4))/((x+5)(x-1)) ≥0

step 1 We have to solve the inequality that x squared multiplied with 3 plus x multiplied with x plus 4

Solve each inequality algebraically....

Key Concepts

Multiplicity of Zeros

Multiplicity refers to the number of times a particular factor appears in a polynomial. This concept is important because it affects the behavior of the graph at the corresponding zero. When a factor has an even multiplicity, the expression does not change sign at that zero; for an odd multiplicity, the sign of the expression changes when passing through the zero. Understanding multiplicity helps to accurately determine the sign pattern required to solve the inequality.

Sign Chart

A sign chart is a systematic method used to determine the sign (positive or negative) of an expression within different intervals. After identifying the critical points, the number line is divided into regions, and a test value is chosen from each interval. The sign of each factor of the expression is then determined in these intervals, and by combining these signs, the overall sign of the expression is computed in each interval.

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials. Solving these inequalities requires finding where the expression is equal to zero (the zeros of the numerator) and where it is undefined (the zeros of the denominator). The solution typically involves determining the sign of the expression over different intervals, taking into account the behavior at the points where the expression is discontinuous.

Critical Points

Critical points are the values of the variable where the expression equals zero or is undefined. These include the roots of the numerator (where the whole expression can be zero) and the roots of the denominator (where the expression is not defined). They are essential to break up the number line into intervals for testing the sign of the inequality.

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