Solve the equation for all values of x. |2x-6|=4x Answer: x=

Absolute Value Equation Split: We have the equation |2x - 6| = 4x. 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 have 2x - 6 = 4x. Now, let's solve for x. Subtract 2x from both sides to get -6 = 2x. Divide both sides by 2 to isolate x. x = -3Positive Case Solution: For the negative case, we have -(2x - 6) = 4x. First, distribute the negative sign inside the parentheses. -2x + 6 = 4x Now, let's solve for x. Add 2x to both sides to get 6 = 6x. Divide both sides by 6 to isolate x. x = 1Negative Case Solution: We need to check if the solutions satisfy the original equation because when dealing with absolute values, extraneous solutions can occur. First, let's check x = -3. Substitute -3 into the original equation |2x - 6| = 4x. |2(-3) - 6| = 4(-3) |-6 - 6| = -12 | -12 | = -12 12 ≠ -12 This means x = -3 is not a solution.Check Solution - x = -3: Now, let's check x = 1. Substitute 1 into the original equation |2x - 6| = 4x. |2(1) - 6| = 4(1) |2 - 6| = 4 |-4| = 4 4 = 4 This means x = 1 is a solution.

Home

Math Problems

Grade 8

Evaluate absolute value expressions

Full solution

Q. Solve the equation for all values of

x

\newline

|2 x-6|=4 x

\newline

Answer:

x=

Absolute Value Equation Split: We have the equation $|2x - 6| = 4x$ . The absolute value equation can be split into two separate equations, one for the positive case and one for the negative case. $\newline$ For the positive case, we have $2x - 6 = 4x$ . $\newline$ Now, let's solve for $x$ . $\newline$ Subtract $2x$ from both sides to get $-6 = 2x$ . $\newline$ Divide both sides by $2$ to isolate $x$ . $\newline$ $x = -3$
Positive Case Solution: For the negative case, we have $- (2x - 6) = 4x$ . $\newline$ First, distribute the negative sign inside the parentheses. $\newline$ $-2x + 6 = 4x$ $\newline$ Now, let's solve for $x$ . $\newline$ Add $2x$ to both sides to get $6 = 6x$ . $\newline$ Divide both sides by $6$ to isolate $x$ . $\newline$ $x = 1$
Negative Case Solution: We need to check if the solutions satisfy the original equation because when dealing with absolute values, extraneous solutions can occur. $\newline$ First, let's check $x = -3$ . $\newline$ Substitute $-3$ into the original equation $|2x - 6| = 4x$ . $\newline$ $|2(-3) - 6| = 4(-3)$ $\newline$ $|-6 - 6| = -12$ $\newline$ $| -12 | = -12$ $\newline$ $12 \neq -12$ $\newline$ This means $x = -3$ is not a solution.
Check Solution - $x = -3$ : Now, let's check $x = 1$ . $\newline$ Substitute $1$ into the original equation $|2x - 6| = 4x$ . $\newline$ $|2(1) - 6| = 4(1)$ $\newline$ $|2 - 6| = 4$ $\newline$ $|-4| = 4$ $\newline$ $4 = 4$ $\newline$ This means $x = 1$ is a solution.