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Chapter 11: Problem 24

Solve polynomial inequality and graph the solution set on a real number line. \(x^{2} \leq 2 x+2\)

Short Answer

Expert verified
The solution of given inequality \(x^{2} \leq 2 x+2\) is \(1 - \sqrt{3} \leq x \leq 1 + \sqrt{3}\)

Step by step solution

01

Step 1

Firstly, arrange the inequality to make one side equals to zero. So, subtract \(2x + 2\) from both sides to get \(x^{2} - 2x - 2 \leq 0\)
02

Step 2

For second step, solve the equality \(x^{2} - 2x - 2 = 0\). This will provide potential points on the number line where the sign of the intervals might change. The quadratic formula is used to solve this, which is \(x = {-b \pm \sqrt{b^2-4ac}}/{2a}\). In this equation, \(a = 1\), \(b = -2\), and \(c = -2\). Plug these values into the formula to get the solutions.
03

Step 3

Using the above values, we get \(x = {2 \pm \sqrt{(-2)^2-4(1)(-2)}}/{2(1)}\). Which further simplifies to \(x = 1 \pm \sqrt{3}\). So, \(x = 1- \sqrt{3}\) and \(x = 1+ \sqrt{3}\).
04

Step 4

Since the coefficient of \(x^2\) is positive, the graph of the quadratic equation is a U shape, and we are looking for where the graph is \(\leq 0\), so the solution will be between the zeros \(1 - \sqrt{3} \leq x \leq 1 + \sqrt{3}\)
05

Step 5

The final solution of the inequality can be graphically presented as marking \(x = 1- \sqrt{3}\) and \(x = 1+ \sqrt{3}\) on the x-axis (number line), and filling in the range between these points including endpoints.

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This formula provides the solutions or "roots" of the equation and is given by:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
In this formula, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation, where \(a eq 0\).
The term under the square root, \(b^2 - 4ac\), is called the discriminant. It determines the number and type of roots:
  • If the discriminant is positive, there are two distinct real roots.
  • If it is zero, there is exactly one real root.
  • If negative, there are two complex roots.
In solving \(x^2 - 2x - 2 = 0\) for example, we substitute \(a = 1\), \(b = -2\), and \(c = -2\) into the quadratic formula to find the roots \(x = 1 \pm \sqrt{3}\), giving us potential points for our inequality analysis.
Number Line Graphing
Number line graphing is a visual representation of the solutions of an inequality or an equation. It helps in understanding which intervals satisfy the given mathematical condition by marking critical points on a number line and analyzing intervals between them.
These steps help ensure accurate graphing of a number line:
  • Identify the solution points from the inequality or equation roots. Here, the roots were \(x = 1 - \sqrt{3}\) and \(x = 1 + \sqrt{3}\).
  • Mark these roots as critical points on the number line with appropriate symbols, like dots or circles, indicating whether they are included or not.
  • Test intervals between these critical points to determine where the inequality is true.
  • Shade the section of the number line that satisfies the inequality. In this case, the solution is the interval \([1 - \sqrt{3}, 1 + \sqrt{3}]\).
This method enables a clear visualization of where the polynomial inequality holds true on the number line.
Quadratic Inequality Solution
Solving a quadratic inequality involves finding the set of values for which the inequality holds. We generally follow these steps:
Start by expressing the inequality such that one side is zero. For \(x^2 \leq 2x + 2\), rearrange to \(x^2 - 2x - 2 \leq 0\).
  • Use the quadratic formula to solve the equivalent equation \(x^2 - 2x - 2 = 0\) to find where the expression may change signs.
  • These solutions \(x = 1 \pm \sqrt{3}\) act as boundaries on the number line.
  • Determine the nature of intervals around these points by testing them. Since the parabola opens upwards (U-shape) and includes \(\leq 0\), the area shaded between and including the two roots represents the solution.
  • The final solution, \([1 - \sqrt{3}, 1 + \sqrt{3}]\), is indicated on the number line with solid dots, showing that both boundary values are part of the solution set.
Understanding the shape and behavior of the function aids significantly in finding where the inequality is satisfied.

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Most popular questions from this chapter

Solve each equation by the method of your choice. $$\sqrt{3} x^{2}+6 x+7 \sqrt{3}=0$$

Write a quadratic inequality whose solution set is \([-3,5]\).

Will help you prepare for the material covered in the first section of the next chapter. In exercise, use point plotting to graph the function. Begin by setting up a table of coordinates, selecting integers from \(-3\) to \(3,\) inclusive, for \(x\). \(f(x)=2^{x}\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the square root property to determine the length of a right triangle's side, I don't even bother to list the negative square root.

Divide: \(\left(x^{4}-5 x^{3}+2 x^{2}-6\right) \div(x-3)\) (Section \(5.6,\) Example 2 or Example 5 )
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