Solve the equation for all values of x. |2x+9|=x Answer: x=

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Solve the equation for all values of  x.  |2x+9|=x Answer:  x=

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Consider Non-Negative Case: We have the equation |2x + 9| = x. To solve this, we need to consider two cases because the absolute value function outputs the distance from zero, which is always non-negative. The two cases are when the expression inside the absolute value is non-negative and when it is negative.Solve for x: First, let's consider the case when the expression inside the absolute value is non-negative, which means 2x + 9 >= 0. In this case, the absolute value function does not change the sign of the expression inside it. So we have 2x + 9 = x. Now, let's solve for x. Subtract x from both sides to get x + 9 = 0. Subtract 9 from both sides to get x = -9.Check Validity of Solution: We need to check if our solution from Step 2 satisfies the original condition 2x + 9 >= 0. Substitute x with -9 and check: 2(-9) + 9 = -18 + 9 = -9, which is not greater than or equal to 0. Therefore, x = -9 does not satisfy the condition 2x + 9 >= 0, so it is not a valid solution.Consider Negative Case: Now, let's consider the second case when the expression inside the absolute value is negative, which means 2x + 9 < 0. In this case, the absolute value function changes the sign of the expression inside it. So we have -(2x + 9) = x. Now, let's solve for x. Distribute the negative sign to get -2x - 9 = x. Add 2x to both sides to get -9 = 3x. Divide both sides by 3 to get x = -3.Solve for x: We need to check if our solution from Step 4 satisfies the original condition 2x + 9 < 0. Substitute x with -3 and check: 2(-3) + 9 = -6 + 9 = 3, which is not less than 0. Therefore, x = -3 does not satisfy the condition 2x + 9 < 0, so it is not a valid solution.Check Validity of Solution: Since neither of the solutions from Step 2 and Step 4 satisfy their respective conditions, we conclude that there are no solutions to the equation |2x + 9| = x.

Solve the equation for all values of $x$ . $\newline$ $|2 x+9|=x$ $\newline$ Answer: $x=$

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Solve the equation for all values of $x$ . $\newline$ $|2 x+9|=x$ $\newline$ Answer: $x=$