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Chapter 5: Problem 30

Solve each inequality algebraically. $$x^{4}<9 x^{2}$$

Short Answer

Expert verified
The solution to the inequality is \( x \in (-3, 0) \cup (0, 3) \).

Step by step solution

01

Rewrite the Inequality

Start by rewriting the given inequality. \[ x^4 < 9x^2 \]
02

Move All Terms to One Side

Subtract \(9x^2\) from both sides to set the inequality to zero. \[ x^4 - 9x^2 < 0 \]
03

Factor the Expression

Factor out the common term \(x^2\). \[ x^2(x^2 - 9) < 0 \]Next, notice that \(x^2 - 9\) is a difference of squares, so we can factor it further. \[ x^2(x - 3)(x + 3) < 0 \]
04

Determine Critical Points

Set each factor to zero to find the critical points. \[ x^2 = 0 \]\[ x - 3 = 0 \]\[ x + 3 = 0 \]So, the critical points are: \[ x = 0, x = 3, x = -3 \]
05

Test Intervals

Test the intervals between and beyond the critical points: Intervals to test: \( (-fty, -3), (-3, 0), (0, 3), (3, fty) \)1. Pick \(x = -4\) from \( (-fty, -3) \). \(( -4 )^2(-4 - 3)(-4 + 3) = 16( -7 )( -1 ) > 0 \) (Not a solution)2. Pick \(x = -1\) from \( (-3, 0) \). \(( -1 )^2(-1 - 3)(-1 + 3) = 1( -4 )( 2 ) < 0 \) (A solution)3. Pick \(x = 1\) from \( (0, 3) \). \(( 1 )^2(1 - 3)(1 + 3) = 1( -2 )( 4 ) < 0 \) (A solution)4. Pick \(x = 4\) from \( (3, fty) \). \(( 4 )^2(4 - 3)(4 + 3) = 16( 1 )( 7 ) > 0 \) (Not a solution)
06

State the Solution

Combine the intervals where the inequality is satisfied.\( x \in (-3, 0) \cup (0, 3) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Inequalities
To solve algebraic inequalities, we need to determine the values of the variable that make the inequality true. Unlike equations, inequalities do not always have a single solution; they often have a range of solutions. An example of an inequality is: \[ x^4 - 9x^2 < 0 \] Here, our goal is to find the values of \( x \) that satisfy this inequality.
Factoring Expressions
Factoring is a crucial step in solving polynomial inequalities. By factoring an expression, we can write it as a product of simpler expressions. For instance, for the expression \[ x^4 - 9x^2 \] we can factor out \( x^2 \): \[ x^2(x^2 - 9) < 0 \] Next, notice that \( x^2 - 9 \) is a difference of squares, which can be factored further into: \[ x^2(x - 3)(x + 3) < 0 \] Factoring helps us identify simpler expressions to work with, paving the way for finding critical points.
Critical Points
Critical points are the values of the variable where the expression equals zero. These points help in breaking down the number line into testable intervals. To find the critical points from the factored inequality \[ x^2(x - 3)(x + 3) < 0 \] we set each factor to zero: \[ x^2 = 0 \]\

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Most popular questions from this chapter

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=x^{4}+x^{3}-3 x^{2}-x+2 $$

Solve each equation in the real number system. $$ x^{4}-2 x^{3}+10 x^{2}-18 x+9=0 $$

Make up an inequality that has no solution. Make up one that has exactly one solution.

Find bounds on the real zeros of each polynomial function. $$ f(x)=4 x^{5}-x^{4}+2 x^{3}-2 x^{2}+x-1 $$

Use the Intermediate Value Theorem to show that the functions \(y=x^{3}\) and \(y=1-x^{2}\) intersect somewhere between \(x=0\) and \(x=1\).
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