Absolute Value and Step Functions Homework Answer Key

Absolute value equations are split into two cases (positive and negative), while step functions require you to identify which flat “piece” your x-value falls into before evaluating. Below, we check both against real homework problems so you can verify your own answers in minutes.

How I Use This With Students

I’ve tutored algebra for years, and these two topics cause the same two mistakes every single time.

• Students forget the negative case in absolute value equations.
• Students plug into the wrong “piece” of a step function.

The walkthroughs below target exactly those two failure points.

Absolute Value: Step-by-Step Answer Check

Solving Equations

Split |expression| = number into two equations.

Example: |x – 4| = 9

• Case 1: x – 4 = 9 → x = 13
• Case 2: x – 4 = -9 → x = -5

Trap to avoid: |x + 2| = -6 has no solution. Distance can’t be negative.

Solving Inequalities

The inequality sign changes the shape of your answer.

Inequality Type	Result	Example
|x| < a	Bounded range (AND)	-a < x < a
|x| > a	Two rays (OR)	x < -a or x > a

Graphing

For f(x) = a|x – h| + k:

• (h, k) is the vertex.
• Negative a flips the V upside down.
• The graph is symmetric around x = h.

Step Functions: Step-by-Step Answer Check

Reading the Graph

Look for two things first.

• Open vs. closed circles at each jump point.
• The domain range is listed for each piece.

Example definition:

f(x) = 2, for 0 ≤ x < 1 f(x) = 4, for 1 ≤ x < 2 f(x) = 6, for 2 ≤ x < 3

Evaluating at a Given x

Find the interval containing your x-value first. Then apply only that piece’s rule. Don’t average pieces or guess.

Real-World Applications

These show up constantly on homework sets.

• Parking garage rates
• Shipping cost tiers
• Tax brackets
• Floor and ceiling rounding

Read boundary wording carefully. “Up to and including 2 hours” is not the same as “under 2 hours.”

Worked Inequality Example

Try this one: |2x – 1| < 7.

• Set up the AND statement: -7 < 2x – 1 < 7.
• Add 1 to all three parts: -6 < 2x < 8.
• Divide everything by 2: -3 < x < 4.

Check it by testing x = 0. Plugging in gives |−1| < 7, which is true. Your range is confirmed.

Now try a “greater than” version: |2x – 1| > 7.

• This splits into two separate inequalities instead of one range.
• 2x – 1 > 7 → x > 4
• 2x – 1 < -7 → x < -3

Notice the answer shape flips entirely. A “less than” problem gives one connected interval. A “greater than” problem gives two disconnected rays.

Common Mistakes on This Topic

These are the errors I see most often when reviewing homework answer keys.

• Dropping the negative case. Students solve |x – 4| = 9 and only report x = 13, forgetting x = -5.
• Flipping the inequality direction incorrectly. Some students reverse both AND and OR cases instead of just one.
• Ignoring open circles on step graphs. An open circle means that the exact point is not included in that piece.
• Assuming step functions are continuous. They aren’t. The jump is the entire point of the function.
• Rounding negative decimals the “normal” way. Floor and ceiling behave differently once numbers go negative.

Catching these five habits fixes the majority of point losses on typical homework sets.

Extra Real-World Step Function Practice

Word problems make step functions click faster than pure equations do. Try walking through this scenario.

A parking garage charges $4 for the first hour, then $2 for each additional hour or part of an hour. A visitor parks for 2 hours and 15 minutes.

• The first hour costs $4 flat.
• Any part of a second hour still counts as a full additional hour.
• Any part of a third hour also counts as a full hour, even 15 minutes.
• Total charge: $4 + $2 + $2 = $8.

This is exactly how ceiling functions work in practice. Partial hours always round up to the next whole step, never down.

4-Step Answer-Checking Method

1. Cover the answer. Redo the problem from scratch.
2. Plug your result back into the original equation.
3. Sketch a quick graph to confirm the shape.
4. Double-check every boundary point on step functions, specifically.

Get a Personalized Answer Key

Generic answer keys often don’t match your textbook’s exact numbering. AiSchool24’s Math solver tools let you paste your own problems and get a free, step-by-step solution to check against.

FAQ

Why does |x| = -3 have no solution?
Distance from zero can never be negative.

Is every step function a piecewise function?
Yes. But not every piecewise function is a step function — steps must be flat, constant pieces.

Does a step function only jump upward?
No. It can jump either direction. The instant jump is the defining feature, not its direction.

Is floor the same as rounding down?
For positive numbers, yes. For negatives, floor(-4.7) = -5, since -5 is the largest integer ≤ -4.7.

About The Author

Shoaib Allam

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