Given the function \( f(x)=-0.5|2 x+2|+1 \), for what values of \( x \) is \( f(x)=6 \) ? \( x=6, x=-5 \) \( x=5, x=-5 \) \( x=7, x=-6 \) no solution DONE

To determine for what values of \( x \) the equation \( f(x)=6 \) is satisfied for the function \( f(x)=-0.5|2 x+2|+1 \), we can set up the equation: \[ -0.5|2 x+2|+1=6 \] Subtracting 1 from both sides gives: \[ -0.5|2 x+2|=5 \] Multiplying by -2 (remembering this reverses the inequality) results in: \[ |2 x+2|=-10 \] Since the absolute value cannot equal a negative number, we can conclude that there is no solution for this equation. Thus, the answer is "no solution." Now let's break it down! How cool is it that absolute values can do such things? They’re like the bouncers at a club – only letting in non-negative numbers! And in the world of equations, if you ever end up with a negative from an absolute value, it means the party’s over – no solution here! When dealing with functions involving absolute values, a common mistake is overlooking the properties of absolute values. Ensure to isolate the absolute value expression correctly, and always check the outcomes. Remember, trying to set an absolute value equal to a negative number is an automatic "no entry!"