Question

Solve each inequality algebraically. $$ x^{2}+2 x-86 \geq-23 $$

   Solve each inequality algebraically.
$$
x^{2}+2 x-86 \geq-23
$$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 6, Problem 50

Instant Answer

Solved by Expert Ashley High

03/05/2020

Step 1

We do this by adding 23 to both sides of the inequality: $$ x^{2}+2 x-86+23 \geq 0 $$ This simplifies to: $$ x^{2}+2 x-63 \geq 0 $$ Show more…

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Solve each inequality algebraically. $$ x^{2}+2 x-86 \geq-23 $$

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Key Concepts

Critical Point and Interval Analysis

After determining the zeros by factoring, the number line is divided into intervals. Testing points from each interval allows one to identify where the quadratic expression is positive or negative. This sign analysis is essential in deciding which intervals form the solution set of the original inequality.

Solving Quadratic Inequalities

This concept focuses on finding the set of variable values that satisfy an inequality involving a quadratic expression. Typically, after rewriting the inequality in standard form and factoring, one identifies the zeros and determines the intervals on the number line where the inequality holds by testing representative values.

Rearranging and Simplifying

This concept involves transforming the inequality to a standard form by moving all terms to one side of the inequality. Doing so simplifies the problem by creating a single quadratic expression compared to zero, which sets the stage for further analysis.

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two linear factors. This method is crucial when solving quadratic inequalities because it reveals the critical points where the expression equals zero, enabling a systematic approach to analyze the sign of the expression.

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