SAT Absolute Value Equations Practice Questions (Free + Explanations)

Advanced Math · Absolute Value Equations

SAT Absolute Value Equations Practice Questions (Free + Explanations) | Quiz 3

Question 12345 of 5

Question 1 of 5

The graph of y = ∣ x - 4∣ is shown in the coordinate plane. Which point lies on this graph?

Explanation For a point (x, y) to lie on the graph of y = ∣ x - 4∣, its coordinates must satisfy the equation. Substituting (4, 0) gives y = ∣4 - 4∣ = 0, which matches the point’s y -value. So (4, 0) lies on the graph. Concept summary A point lies on the graph of an absolute value function when its coordinates make the equation true.

Question 2 of 5

The system y = ∣ x - 4∣ and y = 3 has two solutions. What is the sum of the x -coordinates of those solutions?

Explanation Set the two expressions for y equal: ∣ x - 4∣ = 3 . This means x - 4 = 3 or x - 4 = - 3, so x = 7 or x = 1 . The sum of the x -coordinates is 7 + 1 = 8 . Concept summary When a system includes an absolute value equation and a constant value for the same variable, set them equal and solve the two cases created by the absolute value.

Question 3 of 5

The table shows several values of the function f (x) = ∣ x - 6∣ . x 469 f (x) 203 Based on this pattern, what is the value of x if f (x) = 1 and x > 6 ?

Explanation Since f (x) = ∣ x - 6∣, the equation f (x) = 1 means ∣ x - 6∣ = 1 . This gives x = 7 or x = 5 . Because the question says x > 6, the value must be 7 . Concept summary For an equation of the form ∣ x - a ∣ = b, there are usually two solutions: x = a + b and x = a - b . Extra conditions can determine which one to choose.

Question 4 of 5

A value of x is selected so that ∣ x - 4∣ = 3 . Which statement must be true?

Explanation The equation ∣ x - 4∣ = 3 means the distance between x and 4 is 3 . So x can be 4 + 3 = 7 or 4 - 3 = 1 . Therefore, the statement that must be true is that x is either 1 or 7 . Concept summary An equation of the form ∣ x - a ∣ = b means x is b units away from a, so the solutions are a - b and a + b .

Question 5 of 5

If ∣ x - 4∣ = 7, what is the positive value of x ?

Explanation The equation ∣ x - 4∣ = 7 means the distance between x and 4 is 7 . So x - 4 = 7 or x - 4 = - 7, which gives x = 11 or x = - 3 . The positive value is 11 . Concept summary For an equation of the form ∣ x - a ∣ = b, the solutions are x = a + b and x = a - b when b \geq 0 .

Your results

0of 5 correct

Estimated SAT Math band

500-550

Illustrative range from this short quiz—not an official College Board score.

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Your results

1of 5 correct

Estimated SAT Math band

500-550

Illustrative range from this short quiz—not an official College Board score.

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Your results

2of 5 correct

Estimated SAT Math band

600-650

Illustrative range from this short quiz—not an official College Board score.

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Your results

3of 5 correct

Estimated SAT Math band

600-650

Illustrative range from this short quiz—not an official College Board score.

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Your results

4of 5 correct

Estimated SAT Math band

700+

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

Download on the App Store Get it on Google Play

More SAT Math practice

Your results

5of 5 correct

Estimated SAT Math band

700+

Illustrative range from this short quiz—not an official College Board score.

Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.

Download on the App Store Get it on Google Play

More SAT Math practice

Practice outline (enable JavaScript for interactive mode)

Q1. The graph of y = ∣ x − 4∣ is shown in the coordinate plane. Which point lies on this graph?

Q2. The system y = ∣ x − 4∣ and y = 3 has two solutions. What is the sum of the x -coordinates of those solutions?

Q3. The table shows several values of the function f ( x ) = ∣ x − 6∣ . x 4 6 9  f ( x ) 2 0 3   Based on this pattern, what is the value of x if f ( x ) = 1 and x &gt; 6 ?

Q4. A value of x is selected so that ∣ x − 4∣ = 3 . Which statement must be true?

Q5. If ∣ x − 4∣ = 7 , what is the positive value of x ?