Topics / Algebra

Absolute Value & Inequalities

|x| equations, piecewise definitions, inequality solving, AM-GM, optimization by inequality

Absolute Value & InequalitiesAMC 10 Step by Step

Name: Date: September 13, 2026

Algebra · 25 problems have this as their main idea

primary-topic problems (1.9% of all)

What you need to know

$|x-a|$ is the distance from $x$ to $a$ . $|x|=c$ ( $c>0$ ) gives $x=\pm c$ ; $|x|<c\iff -c<x<c$ ; $|x|>c\iff x>c$ or $x<-c$ .
An expression with several absolute values is piecewise linear, with breakpoints where each inner expression is $0$ ; extreme values occur at breakpoints.
AM-GM: for $a,b\ge0$ , $\dfrac{a+b}2\ge\sqrt{ab}$ , with equality iff $a=b$ . A sum with fixed product is minimized, and a product with fixed sum maximized, when the terms are equal.
Multiplying an inequality by a negative reverses it; squaring is safe only when both sides are nonnegative.

How AMC 10 tests it

Problems 4–10: solve $|2x-5|=7$ or $|x-1|+|x-3|=4$ and report the sum or number of solutions.
Problems 8–14: the region $|x|+|y|\le c$ is a diamond; find its area or count lattice points.
Problems 10–16: minimize $|x-1|+|x-2|+\dots+|x-n|$ (the median) or an expression like $x+\frac4x$ (AM-GM).
Problems 12–18: count integers $n$ satisfying a chain such as $\sqrt n<\frac k3<\sqrt{n+1}$ by careful bounding.

Standard approaches

Translate one or two absolute values into distances on a number line; the answer is often visible.
Otherwise split into cases at the breakpoints, solve each linear piece, and check that each solution lies in its own interval.
For $|x|$ and $|y|$ together, graph the first quadrant and use symmetry across both axes.
For a maximum or minimum, try AM-GM first (confirm equality is attainable), then completing the square.
For integer counting, isolate the variable, find exact endpoints, and count strictly or weakly inside as the wording requires.

Worked example

What is the area of the region of the plane consisting of all points $(x,y)$ with $|x-y|+|x+y|\le4$ ?

(A) $8$ (B) $8\sqrt2$ (C) $16$ (D) $16\sqrt2$ (E) $32$

In the wedge $x\ge|y|$ both $x-y$ and $x+y$ are nonnegative, so the condition is $(x-y)+(x+y)\le4$ , that is, $x\le2$ . That piece is the triangle with vertices $(0,0)$ , $(2,2)$ , $(2,-2)$ , of area $\frac12\cdot2\cdot4=4$ . By symmetry the wedges $x\le-|y|$ , $y\ge|x|$ , $y\le-|x|$ contribute three congruent triangles, so the area is $16$ . (Equivalently, $|x-y|+|x+y|=2\max(|x|,|y|)$ , a square.) The answer is $\boxed{\textbf{(C)}\ 16}$ .

Pitfalls

Accepting a case solution that does not lie in that case's interval.
Solving $|x|>c$ as $-c<x<c$ , confusing "or" with "and".
Applying AM-GM to quantities that may be negative, or not checking that equality is achievable.
Multiplying an inequality by a variable of unknown sign without splitting into cases.

Traps that recur

Reporting the area of one quadrant's diamond (2) without multiplying by 4 for the four quadrants. (2023 AMC 10B #13)
Assuming the parabola crosses the inner diamond's edges in extra points, or miscounting the tangency-type contacts at the vertices (0,3), (3,0), (-3,0). (2021 AMC Fall 10A #14)
Forgetting the 100 integers where P(n) = 0 and answering 5000, or choosing the gaps with even k instead of odd k. (2020 AMC 10A #17)
Removing the outer absolute value as |x| - |y| = 1 only, missing the branch |y| - |x| = 1 that produces the third solution (0, 1). (2018 AMC 10A #12)

Problems, easiest first

2024 AMC 10B · #3Absolute Value & Inequalities

For how many integer values of $x$ is $|2x| \leq 7 \pi$

2021 AMC 10B · #1Absolute Value & Inequalities

How many integer values of $x$ satisfy $|x| < 3\pi$ ?

2018 AMC 10A · #5Absolute Value & Inequalities

Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least $6$ miles away," Bob replied, "We are at most $5$ miles away." Charlie then remarked, "Actually the nearest town is at most $4$ miles away." It turned out that none of the three statements were …

2017 AMC 10B · #3Absolute Value & Inequalities

Real numbers $x$ , $y$ , and $z$ satisfy the inequalities $0<x<1$ , $-1<y<0$ , and $1<z<2$ . Which of the following numbers is necessarily positive?

2016 AMC 10B · #3Absolute Value & Inequalities

Let $x=-2016$ . What is the value of $\bigg| \big||x|-x\big|-|x| \bigg| -x?$

2012 AMC 10B · #6Absolute Value & Inequalities

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$ , Xiaoxi rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

2012 AMC 10A · #3Absolute Value & Inequalities

A bug crawls along a number line, starting at $-2$ . It crawls to $-6$ , then turns around and crawls to $5$ . How many units does the bug crawl altogether?

2011 AMC 10B · #3Absolute Value & Inequalities

At a store, when a length is reported as $x$ inches that means the length is at least $x - 0.5$ inches and at most $x + 0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle?

2011 AMC 10A · #7Absolute Value & Inequalities

Which of the following equations does NOT have a solution?

2004 AMC 10A · #4Absolute Value & Inequalities

What is the value of $x$ if $|x-1|=|x-2|$ ?

2003 AMC 10A · #6Absolute Value & Inequalities

Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$ . Which of the following statements is not true?

2000 AMC 10 · #9Absolute Value & Inequalities

If $|x - 2| = p$ , where $x < 2$ , then $x - p =$

2022 AMC 10A · #6Absolute Value & Inequalities

Which expression is equal to $\left|a-2-\sqrt{(a-1)^2}\right|$ for $a<0?$

2020 AMC 10A · #5Absolute Value & Inequalities

What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

2014 AMC 10A · #7Absolute Value & Inequalities

Nonzero real numbers $x$ , $y$ , $a$ , and $b$ satisfy $x < a$ and $y < b$ . How many of the following inequalities must be true? $\textbf{(I)}\ x+y < a+b\qquad$ $\textbf{(II)}\ x-y < a-b\qquad$ $\textbf{(III)}\ xy < ab\qquad$ $\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}$

2012 AMC 10B · #8Absolute Value & Inequalities

What is the sum of all integer solutions to $1<(x-2)^2<25$ ?

2010 AMC 10A · #11Absolute Value & Inequalities

The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$ . What is $b - a$ ?

2023 AMC 10B · #13Absolute Value & Inequalities

What is the area of the region in the coordinate plane defined by $| | x | - 1 | + | | y | - 1 | \le 1?$

2021 AMC Fall 10A · #14Absolute Value & Inequalities

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align}

2020 AMC 10A · #17Absolute Value & Inequalities

Define $P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).$ How many integers $n$ are there such that $P(n)\leq 0$ ?

2018 AMC 10A · #12Absolute Value & Inequalities

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \begin{align} x+3y&=3 \\ \big||x|-|y|\big|&=1 \end{align}

2011 AMC 10B · #19Absolute Value & Inequalities

What is the product of all the roots of the equation $\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.$

2010 AMC 10B · #13Absolute Value & Inequalities

What is the sum of all the solutions of $x = \left|2x-|60-2x|\right|$ ?

2009 AMC 10A · #16Absolute Value & Inequalities

Let $a$ , $b$ , $c$ , and $d$ be real numbers with $|a-b|=2$ , $|b-c|=3$ , and $|c-d|=4$ . What is the sum of all possible values of $|a-d|$ ?

2004 AMC 10A · #15Absolute Value & Inequalities

Given that $-4\leq x\leq-2$ and $2\leq y\leq4$ , what is the largest possible value of $\frac{x+y}{x}$ ?