Polynomial and Rational Inequalities

Up to this point we have been solving quadratic inequalities.  The technique involving sign charts extends to solving polynomial inequalities of higher degree.

Solving Quadratic, Rational, and Polynomial Inequalities Playlist

Solve.

   Step 1: Determine the critical numbers, which are the roots or zeros in the case of a polynomial inequality.

   Step 2: Create a sign chart.

   Step 3: Use the sign chart to answer the question.

 

 

The last problem shows that not all sign charts will alternate. Do not take any shortcuts and test each interval.

Rational inequalities are solved using the same technique.  The only difference is in the critical numbers.  It turns out that the y-values may change from positive to negative at a restriction. So we will include the zeros of the denominator in our list of critical numbers.

Solve.

 

Tip: Always use open dots for critical numbers that are also zeros of the denominator, or restrictions. This reminds us that they are restrictions and should not be included in the solution set even if the inequality is inclusive.

 

 

 

Use open dots for all of the critical numbers when a strict inequality is involved.

 

 

  

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Absolute Value Equations

Absolute value, by definition, is the distance from zero on a number line.

How to Solve Absolute Value Equations Playlist

If we have |x| = 3 then the question is, “what number can x be so that the distance to zero is 3?” There are two solutions to this question -3 and 3.

In general,

Here n is a positive integer and X represents an algebraic expression called the argument of the absolute value. To solve these equations, set the argument equal to plus or minus n and solve the resulting equations.

Solve.

This technique requires us to first isolate the absolute value. Apply the usual steps for solving to obtain the absolute value alone on one side of the equation, and then set the argument to plus or minus n.

Solve.

When two absolute values are equal we can set the argument of one equal to the argument of the other.

Tip: When in doubt of a solution, check to see if it solves the original equation.  For example, check that {-7, 3} is the solution set for,

Notice that both numbers solve the original equation and therefore we have verified they are in the solution set.

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