Solve each inequality algebraically and write any solution in interval notation. -x^2-4 x-5<0

Name: Problem 388, Chapter 9: Intermediate Algebra
Uploaded: 2022-09-30T16:12:03-08:00
Channel: Amelia Hardy
Description: Solve each inequality algebraically and write any solution in interval notation. -x^2-4 x-5<0

step 1 Here I have the inequality, negative x squared minus 4x minus 5 is less than zero. And I want to

Solve each inequality algebraically and write any solution...

Key Concepts

Interval Notation

Interval notation is a concise way to represent sets of numbers, especially solutions to inequalities. It employs brackets and parentheses to indicate whether endpoints are included or excluded from the solution set.

Critical Points

Critical points are the roots of the corresponding quadratic equation, obtained by setting the quadratic expression equal to zero. These points split the number line into distinct intervals and are essential in determining where the inequality holds by serving as boundaries for sign analysis.

Sign Analysis

Sign analysis, or the test interval method, involves evaluating the sign of the quadratic expression within each interval defined by the critical points. This method helps identify which intervals satisfy the original inequality condition.

Quadratic Inequality

A quadratic inequality involves a quadratic expression set in relation to zero (or another value) using inequality symbols. Solving such inequalities requires determining where the quadratic function takes positive or negative values, which often leads to understanding its overall behavior.

Reversing the Inequality Sign

When an inequality is multiplied or divided by a negative number, it is crucial to reverse the inequality sign. This property ensures that the inequality remains valid after the operation and is particularly important when dealing with quadratic expressions with a negative leading coefficient.

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