How do you solve and find the extraneous solutions for #2\sqrt{43x}+3=0#?
(Notice that it is now clear that this equation has no real solution since the lefthand side cannot be negative.)
by squaring,
which is its extraneous solution.
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To solve the equation 2√(43x) + 3 = 0 and find any extraneous solutions, follow these steps:

Subtract 3 from both sides of the equation: 2√(43x) = 3.

Divide both sides by 2: √(43x) = 3/2.

Square both sides of the equation to eliminate the square root: 43x = (3/2)^2.

Simplify the right side: 43x = 9/4.

Move the constant term to the other side: 3x = 9/4  4.

Simplify the right side: 3x = 9/4  16/4.

Combine the fractions: 3x = 7/4.

Divide both sides by 3 to solve for x: x = (7/4) / 3.

Simplify the right side: x = 7/12.
Therefore, the solution to the equation is x = 7/12. There are no extraneous solutions in this case.
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To solve the equation ( 2\sqrt{43x} + 3 = 0 ) and find any extraneous solutions, follow these steps:

Start by isolating the square root term on one side of the equation: ( 2\sqrt{43x} = 3 )

Square both sides of the equation to eliminate the square root: ( (2\sqrt{43x})^2 = (3)^2 ) ( 4(43x) = 9 )

Expand and simplify the equation: ( 16  12x = 9 )

Move all the terms involving x to one side of the equation: ( 12x = 9  16 ) ( 12x = 7 )

Divide both sides by 12 to solve for x: ( x = \frac{7}{12} ) ( x = \frac{7}{12} )

Check the solution for extraneous solutions by substituting it back into the original equation: ( 2\sqrt{43(\frac{7}{12})} + 3 = 0 ) ( 2\sqrt{4  \frac{7}{4}} + 3 = 0 ) ( 2\sqrt{\frac{16}{4}  \frac{7}{4}} + 3 = 0 ) ( 2\sqrt{\frac{9}{4}} + 3 = 0 ) ( 2(\frac{3}{2}) + 3 = 0 ) ( 3 + 3 = 0 ) ( 6 \neq 0 )
Since the substituted solution ( x = \frac{7}{12} ) does not satisfy the original equation ( 2\sqrt{43x} + 3 = 0 ), there are no real solutions to the original equation, and there are no extraneous solutions to consider.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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