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Mobile App Chapter 5: Problem 23
For which values of \(x\) is \(2 x^2-3 x\) greater than \(x\) ?
A. \(1>x>3\)
B. \(0>x>2\)
C. \(-4
Short Answer
Expert verified
None
Step by step solution
01
- Set Up the Inequality
We start by setting up the inequality given: \[2x^2 - 3x > x\]Then, move all terms to one side of the inequality: \[2x^2 - 3x - x > 0\]This simplifies to: \[2x^2 - 4x > 0\]
02
- Factoring the Quadratic Expression
Next, factor out the common term from the inequality:\[2x(x - 2) > 0\]
03
- Find the Critical Points
To identify the intervals to test, find the roots of the equation:\[2x(x - 2) = 0\]This gives the critical points: \[x = 0\] and \[x = 2\]
04
- Test Intervals Around Critical Points
Divide the number line into regions separated by the critical points and test points in each region:- Interval 1: \(x < 0\)- Interval 2: \(0 < x < 2\)- Interval 3: \(x > 2\)Testing points in each interval (e.g., \(x = -1\), \(x = 1\), \(x = 3\)):\[2(-1)(-1 - 2) > 0\] gives a negative number.\[2(1)(1 - 2) > 0\] gives a negative number.\[2(3)(3 - 2) > 0\] gives a positive number.
05
- Determine Valid Intervals
Based on the tests, the expression \(2x(x - 2) > 0\) holds true for:- \(x > 2\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Inequalities
In quadratic inequalities, the goal is to determine the ranges of values for the variable that make the inequality true. Unlike quadratic equations which have specific roots or solutions, quadratic inequalities often have ranges of solutions.
These solutions are found by transforming the inequality into a standard form where we can easily identify the critical points. The standard form is typically written as:
These solutions are found by transforming the inequality into a standard form where we can easily identify the critical points. The standard form is typically written as:
- \[ax^2 + bx + c < 0\]
- \[ax^2 + bx + c > 0\]
- \[ax^2 + bx + c \, \text{≤} \, 0\]
- \[ax^2 + bx + c \, \text{≥} \, 0\]
Critical Points in Inequalities
Critical points are values of the variable where the expression changes from positive to negative or vice versa. For quadratic inequalities, these points correspond to the roots of the associated quadratic equation.
Consider the inequality \(2x(x - 2) > 0\). The critical points here are the solutions to the equation \(2x(x - 2) = 0\), which gives us \(x = 0\) and \(x = 2\).
We divide the number line into intervals around these critical points to determine the sign of the expression in each interval. By testing points from each interval, we can see whether the inequality holds.
Consider the inequality \(2x(x - 2) > 0\). The critical points here are the solutions to the equation \(2x(x - 2) = 0\), which gives us \(x = 0\) and \(x = 2\).
We divide the number line into intervals around these critical points to determine the sign of the expression in each interval. By testing points from each interval, we can see whether the inequality holds.
- Interval 1: \(x < 0\)
Test with \(x = -1\): \(2(-1)(-3) > 0\) gives negative. - Interval 2: \(0 < x < 2\)
Test with \(x = 1\): \(2(1)(-1) > 0\) gives negative. - Interval 3: \(x > 2\)
Test with \(x = 3\): \(2(3)(1) > 0\) gives positive.
Factoring Quadratic Expressions
Factoring quadratic expressions is a key step in solving quadratic inequalities. This involves expressing the quadratic expression as a product of linear factors.
The quadratic expression from the problem \(2x^2 - 4x\) can be factored by taking out the greatest common factor (GCF): \(2x(x - 2)\).
After factoring, the inequality \(2x^2 - 4x > 0\) becomes \(2x(x - 2) > 0\). This simpler form makes it easier to identify and analyze the critical points and intervals.
Let's summarize the steps to factor a quadratic:
The quadratic expression from the problem \(2x^2 - 4x\) can be factored by taking out the greatest common factor (GCF): \(2x(x - 2)\).
After factoring, the inequality \(2x^2 - 4x > 0\) becomes \(2x(x - 2) > 0\). This simpler form makes it easier to identify and analyze the critical points and intervals.
Let's summarize the steps to factor a quadratic:
- Identify the GCF of the terms if one exists.
- Rewrite the quadratic expression as the product of its factors.
- Use these factors to set up intervals for testing the inequality.
