SAT Nonlinear Equations & Systems Question 121 | Free SAT Prep | Aniko

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Question 121·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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$|2x+7| = 13$ What is the negative solution to the given equation?

For absolute value equations of the form $|ax+b|=k$ with $k>0$ , immediately split them into two linear equations: $ax+b=k$ and $ax+b=-k$ . Solve both quickly using basic algebra, then check what the question actually wants (for example, the negative solution, the larger solution, or all solutions). A brief mental check by plugging each solution back into the absolute value expression helps you catch any sign mistakes.

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Use the definition of absolute value

Remember that if $|A|=k$ (where $k$ is positive), then $A$ can be either $k$ or $-k$ . Apply this idea to $|2x+7|=13$ .

Write and solve two equations

Set $2x+7=13$ and $2x+7=-13$ , then solve each equation for $x$ separately.

Answer the exact question being asked

You will get two values of $x$ . Identify which of those values is negative, since the problem asks specifically for the negative solution.

Desmos Guide

Enter the absolute value expression

In Desmos, type y = abs(2x+7) to graph the function $y=|2x+7|$ .

Graph the constant value

On a new line, type y = 13 to graph the horizontal line $y=13$ .

Find the intersection points

Use Desmos to tap/click the intersection points of the V-shaped graph and the horizontal line. You will see two $x$ -values; identify the one that is negative, since the question asks for the negative solution.

Step-by-step Explanation

Rewrite the absolute value equation without the bars

An equation of the form $|A|=k$ (with $k>0$ ) is equivalent to two equations:

$A=k$
$A=-k$

Here $A=2x+7$ and $k=13$ , so write two separate equations:

$2x+7=13$
$2x+7=-13$

Solve the first linear equation

Solve $2x+7=13$ :

\begin{aligned} 2x+7&=13 \\ 2x&=13-7 \\ 2x&=6 \\ x&=3 \end{aligned}

So one solution of the original equation is $x=3$ .

Solve the second linear equation

Now solve $2x+7=-13$ :

\begin{aligned} 2x+7&=-13 \\ 2x&=-13-7 \\ 2x&=-20 \\ x&=-10 \end{aligned}

So the other solution of the original equation is $x=-10$ .

Choose the negative solution

The question specifically asks for the negative solution. From the two solutions, $x=3$ and $x=-10$ , the negative one is $x=-10$ , which corresponds to answer choice C.

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Step-by-step Explanation

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Step-by-step Explanation