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Chapter 10: Problem 1

Fill in the blanks. \(x^{2}+3 x-18<0\) is an example of a _____ inequality in one variable.

Short Answer

Expert verified
quadratic

Step by step solution

01

Identify the Type of Expression

The given expression is \(x^2 + 3x - 18<0\). This is a polynomial inequality since it involves a polynomial on one side and 0 on the other side.
02

Determine the Degree

The expression \(x^2 + 3x - 18\) is a quadratic polynomial because the highest power of \(x\) is 2. Thus, this inequality is quadratic in nature.

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Key Concepts

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

Polynomial Inequality
Polynomial inequalities involve expressions where a polynomial is compared to another expression, typically zero, using inequality signs such as <, >, 鈮�, or 鈮�. They are used to determine ranges of values that satisfy the inequality condition. In our example, the expression \(x^2 + 3x - 18 < 0\) represents a polynomial inequality because we have a quadratic polynomial on the left-hand side and zero on the right-hand side.

Tackling polynomial inequalities involves several steps:
  • First, rearrange the inequality so that one side is a polynomial and the other side is zero.
  • Solve the corresponding equation \(x^2 + 3x - 18 = 0\) to find the critical points or roots of the polynomial.
  • Analyze the sign changes of the polynomial around these roots to deduce where the inequality holds true.
This process allows us to pinpoint intervals on the number line where the inequality is satisfied.
Inequality in One Variable
An inequality in one variable means that we only have one unknown quantity 鈥� represented with a variable like \(x\) 鈥� whose value we are trying to determine within a specific range. The inequality \(x^2 + 3x - 18 < 0\) is a perfect example. Here, we are looking for all possible values of \(x\) that satisfy this condition.

Working with inequalities in one variable often involves:
  • Isolating the variable on one side, if applicable.
  • Solving the corresponding equations to find any critical values.
  • Testing intervals determined by these critical values to see if they satisfy the inequality.
By focusing on just one variable, these inequalities simplify the process of finding and graphing solution sets on a one-dimensional number line.
Quadratic Polynomial
A quadratic polynomial is an expression that includes a variable raised to the power of two, usually in the format \(ax^2 + bx + c\). In our specific inequality, \(x^2 + 3x - 18\), the polynomial is quadratic because the highest degree of \(x\) is 2.

Key characteristics of quadratic polynomials include:
  • Their graphs form a parabola on a coordinate plane.
  • They can have zero, one, or two real roots depending on the discriminant \(b^2 - 4ac\).
  • Simplifying and factoring a quadratic polynomial can reveal its roots, which are crucial for solving inequalities.
When dealing with a quadratic polynomial inequality, understanding the nature of these roots is essential to finding the intervals where the inequality holds true. Often, recognizing the form of the quadratic can help in deciding the method used to solve it, whether by factoring, completing the square, or using the quadratic formula.

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Most popular questions from this chapter

The height \(h\) (in feet) of an object that is dropped from a height of \(s\) feet is given by the formula \(h=s-16 t^{2}\) where \(t\) is the time the object has been falling. A 5 -foot-tall woman on a sidewalk looks directly overhead and sees a window washer drop a bottle from four stories up. How long does she have to get out of the way? Round to the nearest tenth. (A story is 12 feet.)

Determine the coordinates of the vertex of the graph of each function using the vertex formula. Then determine the \(x-\) and \(y\) -intercepts of the graph. Finally, plot several points and complete the graph. See Example \(9 .\) $$ f(x)=-2 x^{2}+8 x-10 $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ m^{2}-7 m+3=0 $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ (8 x+5)^{2}=24 $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ m^{2}-7 m+3=0 $$
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