Question

write an absolute value equation that represents the graph shown.
your answer should use the form |x - a| = b or |x + a| = b, where a and b are whole numbers, decimals, or simplified fractions.

Explanation:

Step1: Find the midpoint (a)

The two points on the number line are -2 and 5. The midpoint \( a \) is calculated as the average of these two points: \( a=\frac{-2 + 5}{2}=\frac{3}{2}=1.5 \)? Wait, no, wait. Wait, the absolute value equation \(|x - a|=b\) represents the set of points x that are at a distance \( b \) from \( a \). So the two points are -2 and 5. The distance between -2 and 5 is \( 5-(-2)=7 \)? Wait, no, the distance between -2 and 5 is \( |5 - (-2)|=7 \), but the midpoint (the center) is \( \frac{-2 + 5}{2}=\frac{3}{2}=1.5 \)? Wait, no, maybe I made a mistake. Wait, the absolute value equation \(|x - h|=k\) has solutions \( x=h + k \) and \( x=h - k \). So if the two solutions are \( x=-2 \) and \( x = 5 \), then we have:

\( h + k=5 \)

\( h - k=-2 \)

Adding these two equations: \( 2h=3\Rightarrow h = 1.5 \), and then \( k=5 - 1.5 = 3.5 \). But the problem says a and b can be decimals or fractions. Wait, but maybe I misread the points. Wait, the graph has two pink dots at -2 and 5. So the absolute value equation is \(|x - 1.5|=3.5\), but let's check:

\( x - 1.5=3.5\Rightarrow x=5 \)

\( x - 1.5=-3.5\Rightarrow x=-2 \)

Yes, that works. Alternatively, we can write it as \(|x - \frac{3}{2}|=\frac{7}{2}\), but maybe we can use the form \(|x + a|=b\) or \(|x - a|=b\) with decimals. Wait, 1.5 is 3/2, 3.5 is 7/2. But let's see, another way: the midpoint between -2 and 5 is \( \frac{-2 + 5}{2}=1.5 \), and the distance from the midpoint to each point is \( 5 - 1.5 = 3.5 \), so the equation is \(|x - 1.5|=3.5\). Alternatively, we can rewrite \(|x - 1.5|=3.5\) as \(|x - \frac{3}{2}|=\frac{7}{2}\), but maybe we can also write it as \(|x - 1.5| = 3.5\).

Wait, but let's check the problem statement again. It says "your answer should use the form \(|x - a| = b\) or \(|x + a| = b\), where a and b are whole numbers, decimals, or simplified fractions." So 1.5 and 3.5 are decimals, so that's acceptable.

Alternatively, let's solve for h and k:

We know that for \(|x - h|=k\), the solutions are \( x = h + k \) and \( x = h - k \). So we have:

\( h + k = 5 \)

\( h - k = -2 \)

Adding the two equations: \( 2h = 3 \Rightarrow h = 1.5 \)

Subtracting the second equation from the first: \( 2k = 7 \Rightarrow k = 3.5 \)

So the equation is \(|x - 1.5| = 3.5\)

Answer:

\(|x - 1.5| = 3.5\) (or equivalently \(|x - \frac{3}{2}|=\frac{7}{2}\))