Find the $x$-intercept.
$y=\frac{4 x+12}{4 x-12} $
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Answer (2)
Set y = 0 in the equation y = 4 x − 12 4 x + 12 .
Solve the equation 0 = 4 x − 12 4 x + 12 by setting the numerator equal to zero: 4 x + 12 = 0 .
Solve for x : x = − 3 .
The x -intercept is ( − 3 , 0 ) , so the answer is − 3 .
Explanation
Understanding the Problem We are given the equation y = 4 x − 12 4 x + 12 and asked to find the x -intercept. The x -intercept is the point where the graph of the equation intersects the x -axis, which means y = 0 .
Setting y = 0 To find the x -intercept, we set y = 0 in the given equation: 0 = 4 x − 12 4 x + 12 A fraction is equal to zero if and only if its numerator is equal to zero. Therefore, we need to solve the equation: 4 x + 12 = 0
Isolating the x term Subtract 12 from both sides of the equation: 4 x = − 12
Solving for x Divide both sides by 4: x = 4 − 12 = − 3
Finding the x-intercept Therefore, the x -intercept is the point ( − 3 , 0 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in economics, if the equation represents the cost function of a product, the x-intercept would indicate the break-even point where the cost is zero. Similarly, in physics, if the equation describes the trajectory of a projectile, the x-intercept would represent the point where the projectile lands. Knowing how to calculate x-intercepts allows us to analyze and interpret these scenarios effectively, making informed decisions based on the mathematical model.
The x -intercept of the equation y = 4 x − 12 4 x + 12 is found by setting y = 0 , leading us to solve 4 x + 12 = 0 . This gives us x = − 3 , so the x -intercept is at the point ( − 3 , 0 ) .
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