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Solve the equation for all values of 
x.

|4x+6|=2x
Answer: 
x=

Solve the equation for all values of x x .\newline4x+6=2x |4 x+6|=2 x \newlineAnswer: x= x=

Full solution

Q. Solve the equation for all values of x x .\newline4x+6=2x |4 x+6|=2 x \newlineAnswer: x= x=
  1. Split Absolute Value: We have the equation 4x+6=2x|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.\newlineFor the positive case, we simply remove the absolute value bars:\newline4x+6=2x4x + 6 = 2x
  2. Positive Case Solution: Now, let's solve the positive case equation by subtracting 2x2x from both sides to isolate the variable xx on one side:\newline4x+62x=2x2x4x + 6 - 2x = 2x - 2x\newline2x+6=02x + 6 = 0
  3. Positive Case Check: Next, we subtract 66 from both sides to solve for xx: \newline2x+66=062x + 6 - 6 = 0 - 6\newline2x=62x = -6
  4. Negative Case Setup: Finally, we divide both sides by 22 to find the value of xx:2x2=62\frac{2x}{2} = \frac{-6}{2}x=3x = -3
  5. Negative Case Solution: Now, we need to check if the solution x=3x = -3 satisfies the original equation 4x+6=2x|4x + 6| = 2x. We substitute xx with 3-3:4(3)+6=2(3)|4(-3) + 6| = 2(-3)12+6=6|-12 + 6| = -66=6|-6| = -6Since the absolute value of a number is always non-negative, 6|-6| is 66, not 6-6. Therefore, x=3x = -3 does not satisfy the original equation.
  6. 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=2x4x + 6 = -2x
  7. No Solutions: Solve the negative case equation by adding 2x2x to both sides to isolate the variable xx on one side:\newline4x+6+2x=2x+2x4x + 6 + 2x = -2x + 2x\newline6x+6=06x + 6 = 0
  8. No Solutions: Solve the negative case equation by adding 2x2x to both sides to isolate the variable xx on one side:\newline4x+6+2x=2x+2x4x + 6 + 2x = -2x + 2x\newline6x+6=06x + 6 = 0 Subtract 66 from both sides to solve for xx:\newline6x+66=066x + 6 - 6 = 0 - 6\newline6x=66x = -6
  9. No Solutions: Solve the negative case equation by adding 2x2x to both sides to isolate the variable xx on one side:\newline4x+6+2x=2x+2x4x + 6 + 2x = -2x + 2x\newline6x+6=06x + 6 = 0Subtract 66 from both sides to solve for xx:\newline6x+66=066x + 6 - 6 = 0 - 6\newline6x=66x = -6Divide both sides by 66 to find the value of xx:\newline6x6=66\frac{6x}{6} = \frac{-6}{6}\newlinex=1x = -1
  10. No Solutions: Solve the negative case equation by adding 2x2x to both sides to isolate the variable xx on one side:\newline4x+6+2x=2x+2x4x + 6 + 2x = -2x + 2x\newline6x+6=06x + 6 = 0Subtract 66 from both sides to solve for xx:\newline6x+66=066x + 6 - 6 = 0 - 6\newline6x=66x = -6Divide both sides by 66 to find the value of xx:\newline6x6=66\frac{6x}{6} = \frac{-6}{6}\newlinex=1x = -1We need to check if the solution x=1x = -1 satisfies the original equation 4x+6=2x|4x + 6| = 2x. We substitute xx with 1-1:\newline4(1)+6=2(1)|4(-1) + 6| = 2(-1)\newline4+6=2|-4 + 6| = -2\newline2=2|2| = -2Since the absolute value of a number is always non-negative, xx00 is xx11, not xx22. Therefore, x=1x = -1 does not satisfy the original equation either.
  11. No Solutions: Solve the negative case equation by adding 2x2x to both sides to isolate the variable xx on one side:\newline4x+6+2x=2x+2x4x + 6 + 2x = -2x + 2x\newline6x+6=06x + 6 = 0Subtract 66 from both sides to solve for xx:\newline6x+66=066x + 6 - 6 = 0 - 6\newline6x=66x = -6Divide both sides by 66 to find the value of xx:\newline6x6=66\frac{6x}{6} = \frac{-6}{6}\newlinex=1x = -1We need to check if the solution x=1x = -1 satisfies the original equation 4x+6=2x|4x + 6| = 2x. We substitute xx with 1-1:\newline4(1)+6=2(1)|4(-1) + 6| = 2(-1)\newline4+6=2|-4 + 6| = -2\newline2=2|2| = -2\newlineSince the absolute value of a number is always non-negative, xx00 is xx11, not xx22. Therefore, x=1x = -1 does not satisfy the original equation either.Since neither xx44 nor x=1x = -1 satisfy the original equation, there are no solutions to the equation 4x+6=2x|4x + 6| = 2x.

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