Question

which statements are true for solving the equation 0.5 - |x - 12| = -0.25? check all that apply. the equation will have no solutions. a good first step for solving the equation is to subtract 0.5 from both sides of the equation. a good first step for solving the equation is to split it into a positive case and a negative case. the positive case of this equation is 0.5 - |x - 12| = 0.25. the negative case of this equation is x - 12 = -0.75. the equation will have only 1 solution

Explanation:

Step1: Isolate the absolute - value term

First, subtract 0.5 from both sides of the equation \(0.5−|x - 12|=-0.25\). We get \(-|x - 12|=-0.25 - 0.5=-0.75\), then \(|x - 12| = 0.75\).

Step2: Split into positive and negative cases

For the absolute - value equation \(|x - 12| = 0.75\), the positive case is \(x-12 = 0.75\) and the negative case is \(x - 12=-0.75\).

• The original equation \(0.5−|x - 12|=-0.25\) has solutions. So, the statement "The equation will have no solutions" is false.
• A good first step is to subtract 0.5 from both sides of the equation, so the statement "A good first step for solving the equation is to subtract 0.5 from both sides of the equation" is true.
• We split the absolute - value equation into positive and negative cases after isolating the absolute - value term, but splitting it right away is not the first step, so "A good first step for solving the equation is to split it into a positive case and a negative case" is false.
• After isolating the absolute - value term \(|x - 12| = 0.75\), the positive case is \(x - 12=0.75\) not \(0.5−|x - 12| = 0.25\), so this statement is false.
• The negative case of \(|x - 12| = 0.75\) is \(x - 12=-0.75\), so this statement is true.
• The equation \(|x - 12| = 0.75\) has two solutions (\(x=12 + 0.75=12.75\) and \(x=12-0.75 = 11.25\)), so the statement "The equation will have only 1 solution" is false.

Answer:

B. A good first step for solving the equation is to subtract 0.5 from both sides of the equation.
E. The negative case of this equation is \(x - 12=-0.75\).