Q. Solve the equation for all values of x.∣4x+6∣=2xAnswer: x=
Split Absolute Value: We have the equation ∣4x+6∣=2x. The absolute value equation can be split into two separate equations, one for the positive case and one for the negative case.For the positive case, we simply remove the absolute value bars:4x+6=2x
Positive Case Solution: Now, let's solve the positive case equation by subtracting 2x from both sides to isolate the variable x on one side:4x+6−2x=2x−2x2x+6=0
Positive Case Check: Next, we subtract 6 from both sides to solve for x: 2x+6−6=0−62x=−6
Negative Case Setup: Finally, we divide both sides by 2 to find the value of x:22x=2−6x=−3
Negative Case Solution: Now, we need to check if the solution x=−3 satisfies the original equation ∣4x+6∣=2x. We substitute x with −3:∣4(−3)+6∣=2(−3)∣−12+6∣=−6∣−6∣=−6Since the absolute value of a number is always non-negative, ∣−6∣ is 6, not −6. Therefore, x=−3 does not satisfy the original equation.
Negative Case Check: Now, let's consider the negative case of the absolute value equation. We set the expression inside the absolute value equal to the negative of the right side: 4x+6=−2x
No Solutions: Solve the negative case equation by adding 2x to both sides to isolate the variable x on one side:4x+6+2x=−2x+2x6x+6=0
No Solutions: Solve the negative case equation by adding 2x to both sides to isolate the variable x on one side:4x+6+2x=−2x+2x6x+6=0 Subtract 6 from both sides to solve for x:6x+6−6=0−66x=−6
No Solutions: Solve the negative case equation by adding 2x to both sides to isolate the variable x on one side:4x+6+2x=−2x+2x6x+6=0Subtract 6 from both sides to solve for x:6x+6−6=0−66x=−6Divide both sides by 6 to find the value of x:66x=6−6x=−1
No Solutions: Solve the negative case equation by adding 2x to both sides to isolate the variable x on one side:4x+6+2x=−2x+2x6x+6=0Subtract 6 from both sides to solve for x:6x+6−6=0−66x=−6Divide both sides by 6 to find the value of x:66x=6−6x=−1We need to check if the solution x=−1 satisfies the original equation ∣4x+6∣=2x. We substitute x with −1:∣4(−1)+6∣=2(−1)∣−4+6∣=−2∣2∣=−2Since the absolute value of a number is always non-negative, x0 is x1, not x2. Therefore, x=−1 does not satisfy the original equation either.
No Solutions: Solve the negative case equation by adding 2x to both sides to isolate the variable x on one side:4x+6+2x=−2x+2x6x+6=0Subtract 6 from both sides to solve for x:6x+6−6=0−66x=−6Divide both sides by 6 to find the value of x:66x=6−6x=−1We need to check if the solution x=−1 satisfies the original equation ∣4x+6∣=2x. We substitute x with −1:∣4(−1)+6∣=2(−1)∣−4+6∣=−2∣2∣=−2Since the absolute value of a number is always non-negative, x0 is x1, not x2. Therefore, x=−1 does not satisfy the original equation either.Since neither x4 nor x=−1 satisfy the original equation, there are no solutions to the equation ∣4x+6∣=2x.
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