(x%2B5)-0-number-line.jpeg "(x-6)(x+5) 0 Number Line")
Solving Inequalities with a Number Line: (x-6)(x+5) > 0
This article will guide you through solving the inequality (x-6)(x+5) > 0 using a number line.
Understanding the Problem
We need to find the values of 'x' that make the expression (x-6)(x+5) greater than zero. This means we're looking for intervals where the expression is positive.
Steps to Solve
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Find the Critical Points: These are the values of 'x' that make the expression equal to zero. Set each factor to zero and solve:
- x - 6 = 0 => x = 6
- x + 5 = 0 => x = -5
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Number Line: Draw a number line and mark the critical points -5 and 6. These points divide the number line into three intervals:
- Interval 1: x < -5
- Interval 2: -5 < x < 6
- Interval 3: x > 6
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Test Points: Choose a test value from each interval and substitute it into the expression (x-6)(x+5).
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Interval 1 (x < -5): Let's try x = -6 (-6 - 6)(-6 + 5) = (-12)(-1) = 12 (Positive)
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Interval 2 (-5 < x < 6): Let's try x = 0 (0 - 6)(0 + 5) = (-6)(5) = -30 (Negative)
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Interval 3 (x > 6): Let's try x = 7 (7 - 6)(7 + 5) = (1)(12) = 12 (Positive)
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Solution: Based on the test points, the expression (x-6)(x+5) is positive in intervals 1 and 3. Since we want the expression to be greater than zero (> 0), the solution is:
x < -5 or x > 6
Representing the Solution on the Number Line
On the number line, mark the critical points (-5 and 6) with open circles (since the inequality is strictly greater than zero). Shade the intervals to the left of -5 and to the right of 6 to represent the solution.
Conclusion
By using a number line and test points, we determined that the solution to the inequality (x-6)(x+5) > 0 is x < -5 or x > 6. This method provides a visual understanding of the solution and is a helpful tool for solving inequalities.