SAT Math — absolute value equations and V-shaped graphs

Absolute value appears by name in the College Board's description of the Advanced Math domain of the digital SAT, both in equations and in the functions whose graphs students interpret. This material covers both sides of the topic.

On the equation side you practice the two-case method for an equation such as |2x − 7| = 11, and the step that must come first: isolating the absolute value. You then learn to predict the number of solutions before solving — two when the absolute value equals a positive number, one when it equals zero, none when it would have to equal a negative number — and to find the constant that forces each case. Harder questions compare two absolute values, |x − 4| = |x + 2|, and solve an equation whose right side contains the variable, where one of the two cases gives a candidate that must be rejected. A context question turns a manufacturing tolerance into the inequality |L − 2.5| ≤ 0.02, the distance meaning of absolute value.

On the graph side you read the vertex of y = |x − h| + k, find where y = −2|x − 1| + 6 crosses the x-axis, write the equation of a V-shaped graph after a translation and a reflection, find the minimum value of an absolute value function, and count the intersections of y = |x − 2| with a line by solving on each branch of the V.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

  • Solve absolute value equations by isolating the absolute value and splitting into two cases
  • Decide whether |A| = c has two, one or no solutions and find constants that force each case
  • Reject candidates that make the non-absolute side negative
  • Read the vertex, intercepts and transformations of y = a|x − h| + k
  • Express a tolerance in context as an absolute value inequality

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: absolute value equations and functions), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.

Sample question

What are all the real solutions of the equation $|x - 4| = |x + 2|$?

See the answer

$1$ only

Two numbers with equal absolute values are equal or opposite. The case $x - 4 = x + 2$ gives $-4 = 2$, which is false, so it adds nothing. The case $x - 4 = -(x + 2)$ gives $2x = 2$ and $x = 1$. Check: $|1 - 4| = 3$ and $|1 + 2| = 3$. Geometrically, $1$ is the point halfway between $4$ and $-2$ on the number line.

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