(x%2B4)-0-on-a-number-line.jpeg "(x-6)(x+4) 0 On A Number Line")
Solving (x-6)(x+4) = 0 on a Number Line
This problem involves finding the values of x that make the equation true. We can solve this by using the Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero.
1. Factor the equation
The equation is already factored for us: (x - 6)(x + 4) = 0
2. Set each factor equal to zero
- x - 6 = 0
- x + 4 = 0
3. Solve for x in each equation
- x = 6
- x = -4
4. Plot the solutions on a number line
We have two solutions, x = 6 and x = -4. Plot these points on a number line.
5. Determine the intervals
The number line is now divided into three intervals:
- x < -4
- -4 < x < 6
- x > 6
6. Test a value in each interval
Choose a test value within each interval and substitute it into the original equation (x-6)(x+4) = 0. If the result is true, the interval is part of the solution. If the result is false, the interval is not part of the solution.
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x < -4: Let's try x = -5. (-5 - 6)(-5 + 4) = (-11)(-1) = 11. This is not equal to 0, so the interval x < -4 is not a solution.
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-4 < x < 6: Let's try x = 0. (0 - 6)(0 + 4) = (-6)(4) = -24. This is not equal to 0, so the interval -4 < x < 6 is not a solution.
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x > 6: Let's try x = 7. (7 - 6)(7 + 4) = (1)(11) = 11. This is not equal to 0, so the interval x > 6 is not a solution.
7. The solution on the number line
The only solutions are the points where the equation equals zero: x = -4 and x = 6. These are represented by closed circles on the number line.
Solution: The solution to (x-6)(x+4) = 0 on a number line is x = -4 and x = 6.
This can be visualized on a number line with closed circles at -4 and 6, indicating that these points are included in the solution.