Absolute Value Equations · Sub-skill drill
Distance Interpretation of Absolute Value
|x − a| represents the distance between x and a on the number line. This interpretation turns many SAT absolute-value questions into a distance question. For example, |x − 5| ≤ 3 asks for all values within distance 3 of 5, which is the interval from 2 to 8. The interpretation is faster than algebra for inequality questions and works particularly well for word problems involving tolerance, error, or distance from a target value.
How this sub-skill is tested on the SAT
|x − a| represents the distance between x and a on the number line. This interpretation turns many SAT absolute-value questions into a distance question. For example, |x − 5| ≤ 3 asks for all values within distance 3 of 5, which is the interval from 2 to 8. The interpretation is faster than algebra for inequality questions and works particularly well for word problems involving tolerance, error, or distance from a target value.
This sub-skill sits inside the broader Absolute Value Equations topic, which is part of the College Board's Heart of Algebra content domain. Heart of Algebra accounts for roughly a third of every SAT Math section. The College Board frames it as the ability to analyze, fluently solve, and create linear equations and inequalities — and to interpret what their solutions mean in context. If you walk into test day weak here, no amount of advanced math fluency will compensate, because Heart of Algebra questions appear in both the calculator and no-calculator modules and are usually front-loaded so they set the tempo for the entire section. Mastery looks like solving a two-step linear equation in under twenty seconds, recognizing parall
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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If |1x - 4| = 17, what is the sum of all values of x that satisfy the equation?
- A 21
- B -13
- C 8
- D 34
Worked solution
Answer: C — 8
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 1x + (-4) = 17 gives x = 21. Case 2: 1x + (-4) = -17 gives x = -13. Sum = 21 + (-13) = 8.
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If |4x - 10| = 2, what is the sum of all values of x that satisfy the equation?
- A 1
- B 2
- C 5
- D 3
Worked solution
Answer: C — 5
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 4x + (-10) = 2 gives x = 3. Case 2: 4x + (-10) = -2 gives x = 2. Sum = 3 + (2) = 5.
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If |4x - 10| = 2, what is the sum of all values of x that satisfy the equation?
- A 1
- B 2
- C 5
- D 3
Worked solution
Answer: C — 5
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 4x + (-10) = 2 gives x = 3. Case 2: 4x + (-10) = -2 gives x = 2. Sum = 3 + (2) = 5.
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If |2x + 2| = 14, what is the sum of all values of x that satisfy the equation?
- A 6
- B -2
- C -8
- D 14
Worked solution
Answer: B — -2
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 2x + (2) = 14 gives x = 6. Case 2: 2x + (2) = -14 gives x = -8. Sum = 6 + (-8) = -2.
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If |1x + 7| = 3, what is the sum of all values of x that satisfy the equation?
- A -4
- B 6
- C -10
- D -14
Worked solution
Answer: D — -14
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 1x + (7) = 3 gives x = -4. Case 2: 1x + (7) = -3 gives x = -10. Sum = -4 + (-10) = -14.
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If |4x + 3| = 15, what is the sum of all values of x that satisfy the equation?
- A -4
- B -1
- C 7
- D 3
Worked solution
Answer: B — -1
An absolute-value equation |u| = c with c > 0 splits into two cases: u = c and u = -c. Case 1: 4x + (3) = 15 gives x = 3. Case 2: 4x + (3) = -15 gives x = -4. Sum = 3 + (-4) = -1.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.