Question

misha found that the equation $-|2x - 10| - 1 = 2$ had two possible solutions: $x = 3.5$ and $x = -6.5$. which explains whether or not her solutions are correct? she is not correct because there are no solutions. she is not correct because there is only one solution: $x = 3.5$. she is correct because both solutions satisfy the equation. she is not correct because she made a sign error.

Explanation:

Step1: Solve the equation \(-|2x - 10| - 1 = 2\)

First, isolate the absolute value term. Add 1 to both sides: \(-|2x - 10| = 2 + 1 = 3\). Then multiply both sides by -1: \(|2x - 10| = - 3\).

Step2: Analyze the absolute value equation

The absolute value of a number is always non - negative (i.e., \(|a|\geq0\) for any real number \(a\)). In the equation \(|2x - 10|=-3\), the right - hand side is - 3, which is negative. Since the absolute value of a number cannot be negative, this equation has no solution.

Now, let's check Misha's solutions:

• For \(x = 3.5\): Substitute \(x = 3.5\) into the left - hand side of the original equation \(-|2x - 10|-1\). First, calculate \(2x-10\): \(2(3.5)-10 = 7 - 10=-3\). Then \(|2x - 10|=|-3| = 3\). Then \(-|2x - 10|-1=-3 - 1=-4

eq2\).

• For \(x=-6.5\): Substitute \(x=-6.5\) into \(2x - 10\): \(2(-6.5)-10=-13 - 10=-23\). Then \(|2x - 10|=|-23| = 23\). Then \(-|2x - 10|-1=-23 - 1=-24

eq2\).

Misha made a sign error when solving the equation. When she isolated the absolute value, she probably forgot to consider the negative sign in front of the absolute value. For example, when she had \(-|2x - 10|=3\), she should have multiplied both sides by - 1 to get \(|2x - 10|=-3\) (which has no solution), but she might have ignored the negative sign and solved \(|2x - 10| = 3\) instead, which would give solutions \(2x-10 = 3\) or \(2x - 10=-3\). Solving \(2x-10 = 3\) gives \(2x=13\), \(x = 6.5\) (not 3.5), and solving \(2x - 10=-3\) gives \(2x = 7\), \(x = 3.5\). But this is the wrong approach because she didn't handle the negative sign in front of the absolute value correctly.

Answer:

She is not correct because she made a sign error.