Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{3}-2 x^{2}-x+2 \geq 0\)

Short answer

The solution to the inequality is \([-1, 1] ∪ [2,∞)\).

Step-by-step solution

Find the Roots

First, the roots of the equation \(x^{3}-2x^{2}-x+2=0\) need to be found. This can be done by factoring or applying any root-finding method. It is observed that for \(x = -1, 1, 2\), the function \(f(x)=0\). Hence, the roots are \(x = -1, 1, 2\).

Test Intervals

Now it is necessary to check the sign of the function in each of the intervals given by the roots. The intervals are \((-∞, -1)\), \((-1, 1)\), \((1, 2)\) and \((2, ∞)\). Choose a representative from each interval and plug it into the function, looking for whether the function is positive or negative in that interval.

Determine the Solution Set

From testing the intervals, the function is negative between \((-∞, -1)\) and \((1, 2)\), and it's positive between \((-1, 1)\) and \((2, ∞)\). The inequality we want to solve is \(f(x) ≥ 0\), so our solution set will be \([-1,1] ∪ [2, ∞)\) because at \(x = -1, 1 \, and \, 2\), \(f(x) = 0, and f(x) > 0\) for \(x\) in \((-1, 1)\) and \((2, ∞)\).

Key concepts

Graphing on the number line

Visualizing mathematical solutions can make complex concepts more understandable. When you solve polynomial inequalities like \(x^3 - 2x^2 - x + 2 \geq 0\), representing the solution set on a number line can be very helpful. First, identify the solutions or intervals where the polynomial inequality holds true. Once identified, mark these intervals on the number line.

For example, in our solution, the intervals are \([-1,1]\) and \([2, \infty)\). Use solid dots to indicate the inclusion of exact numbers, like \(x = -1\), \(x = 1\), and \(x = 2\), where the inequality becomes an equality. Lines or shading between these points show where the inequality is satisfied. This creates a clear visual that helps you quickly see the solution set.

Root-finding methods

Finding the roots of a polynomial is crucial when solving inequalities. For the equation \(x^3 - 2x^2 - x + 2 = 0\), the roots divide the number line into intervals. These indicate where the polynomial changes sign.

Several methods can be used for root-finding:

• Factoring: This involves expressing the polynomial as a product of simpler polynomials. If it's factorable, it can be the quickest method.
• Trial and error: Plugging in potential roots to see if they make the equation zero.
• Graphical methods: Using graphing calculators or software to visually identify roots.

In this example, roots \(x = -1, 1, 2\) can be found by factoring or testing possible roots. These roots define intervals such as \((-∞, -1)\) and \((1, 2)\), crucial for further analysis.

Testing intervals

Once the roots are identified, testing intervals is the next step. This involves checking the sign of the function in each segment created by those roots. It helps determine where the inequality holds true. With roots at \(x = -1, 1,\) and \(2\), our intervals are \((-∞, -1)\), \((-1, 1)\), \((1, 2)\), and \((2, ∞)\).

To test:

• Pick a random number from each interval (like \(-2\), \(0\), \(1.5\), and \(3\)).
• Substitute this number into the polynomial.
• Check the sign of the result (positive or negative).

This process helps us establish where \(x^3 - 2x^2 - x + 2\) is positive or zero. Identifying these areas is key to determining the solution set of the inequality.

Solution sets

The solution set of an inequality consists of all the values of \(x\) that make the inequality true. After testing each interval, you compile these results to form the solution set. In our example, the function \(x^3 - 2x^2 - x + 2\) is non-negative in the intervals \([-1,1]\) and \([2, ∞)\).

To write a solution set:

• Identify where the inequality \(\geq 0\) is satisfied, including points where \(f(x) = 0\).
• Use interval notation to express this clearly. This usually involves brackets: \([a, b]\) for inclusive intervals and \((a, b)\) for open intervals.

In this particular case, the solution set is \([-1, 1] \cup [2, \infty)\). This set accounts for all x-values making our original inequality true.