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Magazine Chapter 8: Problem 16
Solve each inequality. Write the solution set in interval notation and graph it. $$ x^{2}+2 x-8<0 $$
Short Answer
Expert verified
The solution set is \((-4, 2)\).
Step by step solution
01
Understand the Inequality
The inequality we need to solve is quadratic: \( x^2 + 2x - 8 < 0 \). Solving it involves finding the values of \( x \) that make this expression less than zero.
02
Factor the Quadratic Expression
First, we factor the quadratic expression \( x^2 + 2x - 8 \). We look for two numbers that multiply to \(-8\) and add to \(2\). These numbers are \(4\) and \(-2\). So we rewrite the quadratic as \((x + 4)(x - 2) = 0 \).
03
Find Critical Points
Set each factor equal to zero to find the critical points: \((x + 4) = 0 \Rightarrow x = -4\) \((x - 2) = 0 \Rightarrow x = 2\). These critical points divide the number line into intervals.
04
Test Intervals
We need to determine where \((x + 4)(x - 2) < 0\). This requires testing points in the intervals determined by the critical points, which are \((-\infty, -4)\), \((-4, 2)\), and \((2, \infty)\).- For \((-\infty, -4)\), pick \(x = -5\): \((-5+4)(-5-2) = (-1)(-7) = 7 > 0\)- For \((-4, 2)\), pick \(x = 0\): \((0+4)(0-2) = (4)(-2) = -8 < 0\)- For \((2, \infty)\), pick \(x = 3\): \((3+4)(3-2) = (7)(1) = 7 > 0\)The expression is negative in the interval \((-4, 2)\).
05
Write the Solution in Interval Notation
The solution to the inequality \( x^2 + 2x - 8 < 0 \) is the interval where the expression is negative. This is \((-4, 2)\).
06
Graph the Solution Set
To graph \((-4, 2)\), draw a number line. Place open circles at \(x = -4\) and \(x = 2\) to indicate these points are not included in the solution, then shade the region between these points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratic Expressions
Factoring is like breaking down the expression into simpler parts. Consider a quadratic expression of the form \( ax^2 + bx + c \). To factor it, aim to rewrite it as \((x + m)(x + n)\), where \(m\) and \(n\) are numbers that when multiplied together equal \(c\), and when added equals \(b\). In the expression \( x^2 + 2x - 8 \) from the exercise:
- The coefficient of \(x^2\) is \(1\).
- We look for two numbers that multiply to \(-8\) (the constant term) and add up to \(2\), the coefficient of \(x\).
- These numbers are \(4\) and \(-2\), allowing us to factor the expression as \((x + 4)(x - 2)\).
Interval Notation
Once the critical points are found, we need a way to express intervals where the inequality holds. Interval notation is concise and efficient. It uses brackets to indicate which values are included in the solution set:
- Round brackets \((a, b)\) mean \(a\) and \(b\) are not part of the solution.
- Square brackets \([a, b]\) mean \(a\) and \(b\) are included.
Graphing Inequalities
Graphing helps you visualize where on the number line an inequality is true. Start by drawing a horizontal line (the number line). Mark critical points with open or closed circles:
- Open circles for points not included in the solution, as with \((-4, 2)\).
- Shade the region between \(-4\) and \(2\) to indicate where the inequality \( x^2 + 2x - 8 < 0 \) holds true.
