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Chapter 3: Problem 14

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$ 6 x^{2}+x>1 $$

Short Answer

Expert verified
The solution set for the inequality \(6x^2 + x > 1\) in interval notation is \(x \in (-\infty,x_1) \cup (x_2,+\infty)\).

Step by step solution

01

Rewrite the inequality as a quadratic equation

The given inequality is \(6x^2 + x > 1\). The first step is to subtract 1 from both sides of the equation to bring it into a standard quadratic format: \(6x^2 + x - 1 > 0\)
02

Find the roots

To solve for \(x\), use the quadratic formula, \(-b \pm \sqrt{b^2 - 4ac} / 2a\). Here, \(a = 6\), \(b = 1\), and \(c = -1\). Applying these values in the formula, we get the roots \(x_1\) and \(x_2\) as \(x_1=(-1+\sqrt{1+24})/12\) and \(x_2=(-1-\sqrt{1+24})/12\).
03

Check the inequality sign

The inequality is 'greater than', so we look for the values that make the inequality true outside the roots, i.e, \(xx_2\)
04

Express result in interval notation

The interval notation for the inequality is \(x \in (-\infty,x_1) \cup (x_2,+\infty)\)
05

Graph the solution set

On a number line, we pick points that lie in the intervals defined by the roots \(x_1\) and \(x_2\), and shade the region where the inequality holds true. \(-\infty\) to \(x_1\) and \(x_2\) to \(+\infty\) are shaded to denote the solution set.

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Key Concepts

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

Quadratic Equation
At the heart of many algebraic problems lies the quadratic equation, which is a polynomial equation of the second degree. The general form is given by \( ax^2 + bx + c = 0 \) where \( a \) is not zero. Solving these equations can be critical to various math and science problems. It's a U-shaped curve on a graph known as a parabola, and it always has two solutions or roots, which could be real or complex. The exercise \( 6x^2 + x > 1 \) is actually an inequality based on a quadratic equation when rearranged to \( 6x^2 + x - 1 > 0 \) and provides a base for finding a range of values for \( x \) that satisfies the inequality.
Quadratic Formula
When faced with a quadratic equation, the quadratic formula \( -b \pm \sqrt{b^2 - 4ac} / (2a) \) is a reliable tool to find the equation's roots or solutions. This formula derives from the process of completing the square and offers a straightforward method to solve any quadratic equation. For the inequality \( 6x^2 + x - 1 > 0 \) from the textbook, \( a = 6 \) , \( b = 1 \) , and \( c = -1 \) when plugged into the formula, yields the solutions for \( x \) as \( x_1 \) and \( x_2 \) which are essentially the points where the graph of the quadratic equation crosses the \( x \) -axis.
Interval Notation
Interval notation provides a concise way to express the range of values that include the solutions to inequalities. It's like describing a stretch of numbers on the number line where a particular condition holds true. When we write \( x \in (-\infty,x_1) \cup (x_2,+\infty) \) , it signifies that \( x \) can be any number less than \( x_1 \) or greater than \( x_2 \) , thereby representing the solutions to the given inequality. This kind of notation is key for communicating solutions in a precise and clear manner, which is particularly helpful during assessments or when conveying mathematical ideas.
Number Line Graphing
Graphing on a number line is an invaluable visual tool that helps to illustrate the solutions to inequalities. When we graph the inequality \( 6x^2 + x - 1 > 0 \) , we mark the roots found by the quadratic formula and shade the regions representing the solution set. For 'greater than' inequalities, the areas outside the roots are shaded. This helps to visually confirm the solution and provides a meaningful connection between algebraic expressions and their graphical representations which can be particularly valuable for visual learners.
Roots of a Polynomial
The roots of a polynomial are the solutions to the equation when set to zero. They're the x-intercepts of the polynomial's graph and represent the values of \( x \) for which the polynomial equals zero. In our exercise \( 6x^2 + x - 1 > 0 \) , we are particularly interested in identifying these roots because they act as critical boundary points that separate the intervals on the number line where the inequality's solutions reside. Analyzing the roots helps in understanding the behavior of the polynomial's graph and thereby the nature of solutions to inequalities involving the polynomial.

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