Question

Sehyr solved the equation w2−7=11%0D%0A𝑤%0D%0A2%0D%0A−%0D%0A7%0D%0A=%0D%0A11%0D%0A . She seems to have made an error in her work because the solution she found does not work. Observe Sehyr’s work to find her error and correct it to find the solution to the equation.%0D%0A%0D%0ASehyr’s work:%0D%0A%0D%0Aw2−7=11%0D%0A𝑤%0D%0A2%0D%0A−%0D%0A7%0D%0A=%0D%0A11%0D%0A %0D%0A%0D%0Aw2−7+7=11+7%0D%0A𝑤%0D%0A2%0D%0A−%0D%0A7%0D%0A+%0D%0A7%0D%0A=%0D%0A11%0D%0A+%0D%0A7%0D%0A Using the Addition Property of Equality%0D%0A%0D%0Aw22=182%0D%0A𝑤%0D%0A2%0D%0A2%0D%0A=%0D%0A18%0D%0A2%0D%0A Using the Division Property of Equality%0D%0A%0D%0Aw=9%0D%0A𝑤%0D%0A=%0D%0A9%0D%0A %0D%0A%0D%0A(1 point)

Answer 1

Let's carefully analyze Sehyr's work step by step to identify her error and then solve the equation correctly. The original equation is:

\[ w^2 - 7 = 11 \]

Sehyr's Steps:

1. Adding 7 to both sides: \[ w^2 - 7 + 7 = 11 + 7 \] This step is correct. \[ w^2 = 18 \]

2. Solving for w by taking the square root: Sehyr wrote: \[ w^2 = 18 \implies w = 9 \] This is incorrect because when taking the square root of both sides, you must consider both the positive and negative roots. The correct transformation should be: \[ w = \pm \sqrt{18} \]

Correcting the Work:

From \(w^2 = 18\):

3. Taking the square root: \[ w = \sqrt{18} \quad \text{or} \quad w = -\sqrt{18} \]

Since \( \sqrt{18} \) can be simplified: \[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]

Thus, the solutions are: \[ w = 3\sqrt{2} \quad \text{and} \quad w = -3\sqrt{2} \]

Final Solution:

The correct solutions to the equation \(w^2 - 7 = 11\) are: \[ w = 3\sqrt{2}, \quad w = -3\sqrt{2} \]

Sehyr's mistake was omitting the negative root after taking the square root of both sides.