Domain: Advanced Math | Skill: Nonlinear equations in one variable and systems of equations in two variables | Difficulty: Medium

Opening Hook: Why Nonlinear Equations Matter for SAT Math Success

Nonlinear equations are the gateway to higher-scoring SAT Math sections. Unlike their linear counterparts, these equations involve variables raised to powers other than 1, absolute values, or products of variables—making them more challenging but also more rewarding to master. At the medium difficulty level, you’ll encounter quadratic equations, absolute value problems, and systems mixing linear with nonlinear components. These questions test your ability to think flexibly and apply multiple solving strategies, skills that separate solid math students from exceptional ones.

Question Types Table

Typical Format	What It Tests	Quick Strategy
\(|ax + b| = c\)	Understanding absolute value creates two cases	Set up \(ax + b = c\) and \(ax + b = -c\)
\(x^2 + bx + c = 0\)	Quadratic solving via factoring or formula	Check if it factors nicely first
System: linear + quadratic	Substitution and handling multiple solutions	Solve linear for one variable, substitute
\((x – h)^2 = k\)	Square root property application	Take ± square root of both sides

Real SAT-Style Example

Question: \(|5x – 15| = 50\)

What is the positive value of \(x – 3\)?

Answer Choices:

A) 10 ✅

B) -10

C) 13

D) -7

Step-by-Step Solution:

First, we need to solve the absolute value equation \(|5x – 15| = 50\).

An absolute value equation creates two cases:

• Case 1: \(5x – 15 = 50\)
• Case 2: \(5x – 15 = -50\)

Solving Case 1:

\[5x – 15 = 50\] \[5x = 65\] \[x = 13\]

Solving Case 2:

\[5x – 15 = -50\] \[5x = -35\] \[x = -7\]

Now we need to find the positive value of \(x – 3\):

• When \(x = 13\): \(x – 3 = 13 – 3 = 10\)
• When \(x = -7\): \(x – 3 = -7 – 3 = -10\)

The positive value is 10, so the answer is A.

Step-by-Step Strategy

1. Identify the equation type: Determine if you’re dealing with absolute value, quadratic, radical, or a system of equations.
2. Set up the problem correctly: For absolute values, create two cases; for quadratics, get everything to one side; for systems, identify which method (substitution or elimination) works best.
3. Solve systematically: Work through each case or equation carefully, showing all algebraic steps.
4. Check all solutions: Some nonlinear equations produce extraneous solutions that don’t work in the original equation.
5. Answer what’s asked: SAT often asks for expressions involving your solution, not the solution itself.

Applying the Strategy to Our Example

Step 1 applied: We identify this as an absolute value equation: \(|5x – 15| = 50\). This tells us we’ll need to create two separate cases to solve.

Step 2 applied: We set up two cases based on the definition of absolute value:

Case 1: \(5x – 15 = 50\) (when the expression inside is positive)

Case 2: \(5x – 15 = -50\) (when the expression inside is negative)

Step 3 applied: We solve each case systematically:

Case 1: \(5x = 65\), so \(x = 13\)

Case 2: \(5x = -35\), so \(x = -7\)

Both solutions are valid for the absolute value equation.

Step 4 applied: We verify by substituting back:

For \(x = 13\): \(|5(13) – 15| = |65 – 15| = |50| = 50\) ✓

For \(x = -7\): \(|5(-7) – 15| = |-35 – 15| = |-50| = 50\) ✓

Step 5 applied: The question asks for the positive value of \(x – 3\):

When \(x = 13\): \(x – 3 = 10\) (positive)

When \(x = -7\): \(x – 3 = -10\) (negative)

The answer is 10.

Ready to Try It on Real Questions?

Now that you understand the strategy, it’s time to practice with authentic SAT questions! Head to mytestprep.ai and follow these steps:

1 . Login using your account or signup on mytestprep.ai
2 . Click on Practice Sessions once you are on the dashboard. You will see the link on the left side navigation menu of the dashboard
3 . Click on Create New Session
4 . Start with Co-Pilot Mode on with hints and explanations—it’s like having a personal coach who explains exactly why each answer is right or wrong
5 . Select Math as your subject
6 . Select Advanced Math under Domain, Nonlinear equations in one variable and systems of equations in two variables as skill and Medium difficulty
7 . Select desired number of questions
8 . Start practicing. Happy Practicing!

Key Takeaways

• Absolute value equations always create two cases – don’t forget to solve both!
• Check your solutions – nonlinear equations can produce extraneous solutions
• Read carefully – SAT often asks for expressions involving your solution, not the solution itself
• Factor first when possible – it’s usually faster than the quadratic formula
• Practice pattern recognition – medium-level problems often combine concepts you’ve seen before