Question 1012·Easy·Math

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$|5 - x| = 12$

What is the negative solution to the equation above?

An absolute value equation with a positive number outside the bars can have two solutions: the expression inside can equal that number or its opposite. For a fast Desmos solve, use a regression and restrict the unknown to the sign the question requests. The trap is that a negative value inside the bars does not necessarily give a negative solution.

Hints

Hint 1
Absolute value measures distance from zero: both $12$ and $-12$ are $12$ units from zero. So the expression inside the bars can equal either number. What two equations does that give you?
Hint 2
The word negative describes $x$ , not $5-x$ . In Desmos, use $x_1$ for the unknown and restrict it to $x_1<0$ so the regression finds the solution the question asks for.

Step-by-step

Approach 1: Use a restricted Desmos regression

Step 1
Identify the two possible values inside the bars
An absolute value is a distance from zero. Both $12$ and $-12$ have a distance of $12$ , so the given equation means $5-x=12\quad\text{or}\quad 5-x=-12$ . The negative solution refers to $x$ , not to which value is inside the bars. Find the sign of $x$ before choosing a case.
Step 2
Ask Desmos for the negative solution
Type $|5-x_1|\sim12\{x_1<0\}$ . A regression uses $\sim$ to ask Desmos to find a value; write $x_1$ instead of $x$ because Desmos uses $x$ for graphing. The braces give it a negative restriction, so it won't return the positive solution. Under PARAMETERS, Desmos shows $x_1=-7$ . The negative solution is $-7$ . Choice C.

Approach 2: Solve both cases by hand

Step 1
Solve the first case
Start with $5-x=12$ . Subtract $5$ from both sides: $-x=7$ . Multiply by $-1$ : $x=-7$ . The minus on $x$ matters; stopping at $-x=7$ and reporting $7$ would change the solution.
Step 2
Solve the other case and select
For $5-x=-12$ , subtract $5$ from both sides: $-x=-17$ . Multiply by $-1$ : $x=17$ . That is also a solution, but it's positive. The negative solution is $-7$ . Choice C.

Question 1012·Easy·Math

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Strategy

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