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Chapter 0: Problem 4

Describe the set by listing its elements. $$ |m \in Z| m^{2}-m<115 \mid $$

Short Answer

Expert verified
The set is \(-10, -9, -8, \ldots, 11\).

Step by step solution

01

Understand the Set

The set we are dealing with is defined by the condition \( |m \in \mathbb{Z}| \text{ and } m^2 - m < 115| \). This means we are looking for integer values of \( m \) such that \( m^2 - m < 115 \).
02

Simplify the Inequality

Rewrite the inequality as \( m^2 - m - 115 < 0 \). Our goal is to find values of \( m \) that satisfy this inequality.
03

Solve the Quadratic Inequality

To solve the inequality \( m^2 - m - 115 < 0 \), first find the roots of the equation \( m^2 - m - 115 = 0 \) using the quadratic formula: \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \( a = 1 \), \( b = -1 \), and \( c = -115 \).
04

Calculate the Roots

Calculate the roots: \( m = \frac{1 \pm \sqrt{1 + 460}}{2} = \frac{1 \pm \sqrt{461}}{2} \). Since \( \sqrt{461} \approx 21.47 \), the approximate roots are \( m_1 \approx \frac{1 + 21.47}{2} = 11.235 \) and \( m_2 \approx \frac{1 - 21.47}{2} = -10.235 \).
05

Determine the Range for m

Since we need integer values, we evaluate the integers between the roots. The integers between -10.235 and 11.235 are from -10 to 11.
06

Verify the Integer Values

Verify that each integer value of \( m \) from -10 to 11 satisfies the inequality \( m^2 - m < 115 \). Substitute these values and check if the condition holds true.
07

List the Elements of the Set

The integers that satisfy \( m^2 - m < 115 \) are \(-10, -9, -8, \ldots, 10, 11\).

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Key Concepts

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

Integer Solutions
When dealing with quadratic inequalities, one often needs to find integer solutions. These are specific values that satisfy the given conditions and are whole numbers without any fractions or decimals. To find integer solutions, you need to solve the inequality and look for integer values that fit the solution.
  • For the inequality \(m^2 - m < 115\), integer solutions refer to the whole number values of \(m\) that satisfy this relation.
  • In our example, these numbers range from \(-10\) to \(11\).
  • It is important to confirm each integer value within this range meets the inequality condition.
This approach ensures the solution is complete and accurate for the problem.
Quadratic Formula
The quadratic formula is a powerful tool to find the roots of any quadratic equation, which has the standard form \(ax^2 + bx + c = 0\). The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps us find the roots of the equation by solving for the values of \(x\) that make the equation equal to zero.
  • In our case, the quadratic equation is \(m^2 - m - 115 = 0\).
  • Substituting \(a = 1\), \(b = -1\), and \(c = -115\) into the formula helps us find the necessary roots.
  • The roots tell us where the quadratic expression changes from negative to positive or vice versa.
This step is crucial to solve quadratic inequalities effectively.
Quadratic Roots
Quadratic roots are the solutions to the quadratic equation and play a key role in understanding where an expression may be positive or negative.
  • For the equation \(m^2 - m - 115 = 0\), we calculate roots using the quadratic formula.
  • The approximate roots found are \(m_1 \approx 11.235\) and \(m_2 \approx -10.235\).
  • These roots indicate the critical points where the graph of the quadratic equation intersects the x-axis.
Understanding these points allows us to analyze the intervals that satisfy the inequality condition. This knowledge helps simplify finding integer solutions within the specified range.
Inequality Solving
Solving a quadratic inequality involves a few specific steps to determine the correct set of solutions. Here is how you tackle it:
  • First, express the inequality in the form \(m^2 - m - 115 < 0\).
  • Find the roots of the corresponding quadratic equation \(m^2 - m - 115 = 0\) using the quadratic formula.
  • Determine the intervals defined by these roots to see where the inequality holds true.
  • Test integer values within these intervals to ensure they satisfy the inequality condition.
By following these steps, you will be able to find a precise and complete set of integer values that form the solution to the inequality. This systematic method makes solving quadratic inequalities much more manageable.

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