Question

Solve the inequality and write the solution set in interval notation. \(x^{3}-9 x \leq 0\)

Short answer

The solution for the inequality is \([-3,0] \cup [0,3]\].

Step-by-step solution

Simplification

Factor the inequality: \(x^{3}-9x \leq 0\). Factor out \(x\) from each term, giving us, \(x(x^{2}-9) \leq 0\). Further, since the factor \(x^{2}-9\) is a difference of squares, it can be factored into \((x-3)(x+3)\). Thus, the inequality becomes \(x(x-3)(x+3)\leq 0\).

Finding Critical Points

Set the factored inequality to 0, to find the critical points. Critical points are obtained when the factored equation has a product of zero, thus \(x=0\), \(x=3\), and \(x=-3\) .

Constructing a Sign Chart

Draw a number line including the critical points. Pick a test point from each interval, substitute it in the inequality, and determine the sign (positive or negative). The intervals are: \(-\infty , -3\), \(-3 ,0\), \(0 ,3\), and \(3 , \infty\). For example, if we choose \(-4\) from the first interval and substitute it into the inequality, we get a positive result. Repeat this procedure for the other intervals.

Finding the Solution Set

The solution set is where the inequality is less than or equals to zero. From the sign chart, the solution set is \(-3\leq x\leq 0\) and \(0\leq x\leq 3\).

Write Solution in Interval Notation

Interval notation for the solution set is \([-3,0] \cup [0,3]\]. The square brackets imply that the endpoints are included in the solution.

Key concepts

Factoring

Factoring is a method used to simplify mathematical expressions by breaking them down into products of their simpler components. In this inequality, we factor the expression \(x^3 - 9x\) by first identifying common factors. Here, \(x\) is a common factor in each term, allowing us to factor it out, resulting in \(x(x^2 - 9)\). This is only the first step. On further inspection, the expression within the parenthesis, \(x^2 - 9\), is recognized as a difference of squares. Recognizing patterns such as a difference of squares can simplify the expressions further. We can factor \(x^2 - 9\) into \((x - 3)(x + 3)\). Therefore, the original expression \(x^3 - 9x\) is fully factored into \(x(x - 3)(x + 3)\). Factoring is crucial because it can transform a complex expression into a simpler form, making it easier to solve inequalities or equations.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is used to represent the solution set of an inequality conveniently. In the inequality \(x^3 - 9x \leq 0\), after solving, we are left with solutions that define ranges on the number line. For example, this solution includes the intervals \([-3,0]\) and \([0,3]\).

• The square brackets \([\cdot,\cdot]\) indicate that the endpoints are included in the solution, signifying 'equal to' in the inequality \(\leq\).
• Parentheses \((\cdot,\cdot)\) would have been used if the endpoints were exclusive, representing 'less than' or 'greater than'.

Symbols like \(\cup\) denote the union of sets, indicating that the solutions span both intervals \([-3,0]\) and \([0,3]\). With practice, interval notation becomes a streamlined method to write solution sets clearly and succinctly.

Critical Points

Critical points in the context of inequalities are the values where the expression equals zero. These points are pivotal in determining the intervals of a number line to be analyzed. To find them, we set the factored inequality equal to zero, \(x(x-3)(x+3) = 0\). Each point where any of the factors becomes zero is a critical point.

• For \(x = 0\), the factor \(x\) is zero.
• For \(x = -3\), the factor \(x+3\) is zero.
• For \(x = 3\), the factor \(x-3\) is zero.

These critical points divide the number line into intervals which can then be tested for positivity or negativity. Identifying these points is crucial since the solutions to the inequality will depend on the signs of the intervals they define.

Sign Chart

A sign chart is a valuable tool for solving inequalities because it helps determine which intervals on the number line satisfy the inequality. After identifying the critical points, the number line is divided into intervals, which are tested using a sign chart to find where the inequality holds. To construct a sign chart:

• Mark the critical points on a number line.
• Choose a test point from each interval.
• Substitute these points into the factored inequality.
• Record the sign (positive or negative) of the resulting value for each interval.

For example, examining the interval \([-3,0]\) involves selecting a number like \(-1\). Substituting it into \(x(x-3)(x+3)\) shows a negative result. Repeat this step for each identified interval to construct the sign chart.This chart visually and logically maps out solutions, indicating where the product is less than or equal to zero. Such graphical representations simplify the process of deriving which intervals satisfy the inequality.