SAT Nonlinear Equations & Systems Question 37 | Free SAT Prep | Aniko

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Question 37·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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$|5 - x| = 12$

What is the negative solution to the equation above?

When you see an equation with an absolute value, like $|A| = k$ , immediately split it into two linear equations: $A = k$ and $A = -k$ . Solve both equations carefully, watching your signs when you move terms across the equals sign. If the question specifies a positive or negative solution, or asks for how many solutions there are, use your two answers to match exactly what the question requests instead of just picking the first solution you find.

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Use the definition of absolute value

If $|A| = 12$ , what two values can $A$ have? Apply this idea to the expression $5 - x$ .

Set up two equations

Rewrite $|5 - x| = 12$ as two separate equations: one where $5 - x$ equals a positive number and one where it equals the corresponding negative number.

Solve and then choose carefully

After you find both values of $x$ , check which one is negative. That is the one the question is asking for.

Desmos Guide

Graph the absolute value expression

In Desmos, type y = abs(5 - x) to represent the left side of the equation $|5 - x|$ as a graph.

Add the constant value 12

On a new line, type y = 12 to create a horizontal line that represents the right side of the equation.

Find the intersection points

Look for the points where the graph of y = abs(5 - x) intersects the line y = 12. There will be two intersection points; note both $x$ -values.

Select the required solution

From the two $x$ -values you found, choose the one that is negative. That is the value asked for in the question.

Step-by-step Explanation

Understand what absolute value means

For any expression $A$ , the equation $|A| = 12$ means that $A$ can be either $12$ or $-12$ , because both $12$ and $-12$ are 12 units from 0 on the number line.

Here, $A$ is $5 - x$ , so we need to consider both possibilities for $5 - x$ .

Write the two linear equations

From $|5 - x| = 12$ , set up the two cases:

Case 1: $5 - x = 12$
Case 2: $5 - x = -12$

Next, solve each of these equations for $x$ .

Solve each case for x

Solve Case 1:

\begin{aligned} 5 - x &= 12 \\ -x &= 12 - 5 \\ -x &= 7 \end{aligned}

Solve Case 2:

\begin{aligned} 5 - x &= -12 \\ -x &= -12 - 5 \\ -x &= -17 \end{aligned}

These lead to two values of $x$ ; we will choose the required one next.

Choose the negative solution

The solutions to $|5 - x| = 12$ are $x = -7$ and $x = 17$ . The question asks specifically for the negative solution, which is $x = -7$ .

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Step-by-step Explanation

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Step-by-step Explanation