SAT Math
UNIT 1: Algebra (Foundation)
Solving Linear Equations & Inequalities
1. Linear Equations: Core Concepts
Definition:
• A linear equation in one variable is an equation that can
be written as:
ax+b=cax+b=c
o aa, bb, and cc are constants (a≠0a =0).
o xx is the variable to solve for.
Purpose:
• Find the value of xx that makes the equation true.
2. Step-by-Step Solving Process
Step 1: Simplify Both Sides
• Distributive Property: Eliminate parentheses by
distributing coefficients.
a(b+c)=ab+aca(b+c)=ab+ac
Example:
3(2x−4)=6x−123(2x−4)=6x−12
• Combine Like Terms: Group variables and constants
separately.
Example:
2x+5+4x−3=6x+22x+5+4x−3=6x+2
,Step 2: Isolate the Variable
• For Equations:
o Add/Subtract terms to move variables to one side
and constants to the other.
o Multiply/Divide to solve for xx.
Example:
2(x+3)=4x−62(x+3)=4x−6
→ Distribute: 2x+6=4x−62x+6=4x−6
→ Subtract 2x2x: 6=2x−66=2x−6
→ Add 6: 12=2x12=2x
→ Divide by 2: x=6x=6.
• For Inequalities:
o Follow the same steps as equations, but reverse the
inequality sign when multiplying/dividing by a
negative number.
Example:
−3x>9−3x>9
→ Divide by -3 (reverse sign): x<−3x<−3.
Step 3: Verify Solutions
• Equations: Plug xx back into the original equation.
Example:
If x=6x=6,
check 2(6+3)=4(6)−6→18=182(6+3)=4(6)−6→18=18.
• Inequalities: Test values in the solution set and outside
it.
Example: For x<−3x<−3:
o Test x=−4x=−4: −3(−4)=12>9−3(−4)=12>9
, o Test x=0x=0: −3(0)=0≯9−3(0)=0 >9
3. Special Cases & Edge Scenarios
1. No Solution:
o Occurs when simplifying leads to a false statement
(e.g., 0x=50x=5).
Example:
2x+3=2x−52x+3=2x−5
→ Subtract 2x2x: 3=−53=−5 (No solution).
2. Infinite Solutions:
o Occurs when both sides are identical
(e.g., 0x=00x=0).
Example:
4x−6=2(2x−3)4x−6=2(2x−3)
→ Distribute: 4x−6=4x−64x−6=4x−6 (All real numbers are
solutions).
4. Word Problems: Translation & Setup
Example 1 (Equation):
"A taxi charges a 3basefeeplus3basefeeplus2 per mile. You
paid $15. How many miles did you ride?"
• Let x=x= miles.
• Equation: 3+2x=153+2x=15.
• Solve: 2x=12→x=62x=12→x=6.
, Example 2 (Inequality):
"You have 50.T−shirtscost50.T−shirtscost12 each. How many
can you buy without exceeding your budget?"
• Let x=x= number of shirts.
• Inequality: 12x≤5012x≤50.
• Solve: x≤5012≈4.17→x≤4x≤1250≈4.17→x≤4.
5. Common Mistakes & How to Avoid Them
1. Forgetting to Flip Inequality Signs:
o Error: −2x<8→x<−4−2x<8→x<−4.
o Fix: Divide by -2 and flip: x>−4x>−4.
2. Misapplying the Distributive Property:
o Error: 2(x+3)≠2x+32(x+3) =2x+3.
o Fix: 2(x+3)=2x+62(x+3)=2x+6.
3. Sign Errors:
o Error: Solving −x=5→x=5−x=5→x=5.
o Fix: Multiply both sides by -1: x=−5x=−5.
4. Mishandling Fractions:
o Error: Leaving fractions uncleared.
o Fix: Multiply all terms by the denominator.
Example:
12x+3=521x+3=5
→ Multiply by 2: x+6=10→x=4x+6=10→x=4.
UNIT 1: Algebra (Foundation)
Solving Linear Equations & Inequalities
1. Linear Equations: Core Concepts
Definition:
• A linear equation in one variable is an equation that can
be written as:
ax+b=cax+b=c
o aa, bb, and cc are constants (a≠0a =0).
o xx is the variable to solve for.
Purpose:
• Find the value of xx that makes the equation true.
2. Step-by-Step Solving Process
Step 1: Simplify Both Sides
• Distributive Property: Eliminate parentheses by
distributing coefficients.
a(b+c)=ab+aca(b+c)=ab+ac
Example:
3(2x−4)=6x−123(2x−4)=6x−12
• Combine Like Terms: Group variables and constants
separately.
Example:
2x+5+4x−3=6x+22x+5+4x−3=6x+2
,Step 2: Isolate the Variable
• For Equations:
o Add/Subtract terms to move variables to one side
and constants to the other.
o Multiply/Divide to solve for xx.
Example:
2(x+3)=4x−62(x+3)=4x−6
→ Distribute: 2x+6=4x−62x+6=4x−6
→ Subtract 2x2x: 6=2x−66=2x−6
→ Add 6: 12=2x12=2x
→ Divide by 2: x=6x=6.
• For Inequalities:
o Follow the same steps as equations, but reverse the
inequality sign when multiplying/dividing by a
negative number.
Example:
−3x>9−3x>9
→ Divide by -3 (reverse sign): x<−3x<−3.
Step 3: Verify Solutions
• Equations: Plug xx back into the original equation.
Example:
If x=6x=6,
check 2(6+3)=4(6)−6→18=182(6+3)=4(6)−6→18=18.
• Inequalities: Test values in the solution set and outside
it.
Example: For x<−3x<−3:
o Test x=−4x=−4: −3(−4)=12>9−3(−4)=12>9
, o Test x=0x=0: −3(0)=0≯9−3(0)=0 >9
3. Special Cases & Edge Scenarios
1. No Solution:
o Occurs when simplifying leads to a false statement
(e.g., 0x=50x=5).
Example:
2x+3=2x−52x+3=2x−5
→ Subtract 2x2x: 3=−53=−5 (No solution).
2. Infinite Solutions:
o Occurs when both sides are identical
(e.g., 0x=00x=0).
Example:
4x−6=2(2x−3)4x−6=2(2x−3)
→ Distribute: 4x−6=4x−64x−6=4x−6 (All real numbers are
solutions).
4. Word Problems: Translation & Setup
Example 1 (Equation):
"A taxi charges a 3basefeeplus3basefeeplus2 per mile. You
paid $15. How many miles did you ride?"
• Let x=x= miles.
• Equation: 3+2x=153+2x=15.
• Solve: 2x=12→x=62x=12→x=6.
, Example 2 (Inequality):
"You have 50.T−shirtscost50.T−shirtscost12 each. How many
can you buy without exceeding your budget?"
• Let x=x= number of shirts.
• Inequality: 12x≤5012x≤50.
• Solve: x≤5012≈4.17→x≤4x≤1250≈4.17→x≤4.
5. Common Mistakes & How to Avoid Them
1. Forgetting to Flip Inequality Signs:
o Error: −2x<8→x<−4−2x<8→x<−4.
o Fix: Divide by -2 and flip: x>−4x>−4.
2. Misapplying the Distributive Property:
o Error: 2(x+3)≠2x+32(x+3) =2x+3.
o Fix: 2(x+3)=2x+62(x+3)=2x+6.
3. Sign Errors:
o Error: Solving −x=5→x=5−x=5→x=5.
o Fix: Multiply both sides by -1: x=−5x=−5.
4. Mishandling Fractions:
o Error: Leaving fractions uncleared.
o Fix: Multiply all terms by the denominator.
Example:
12x+3=521x+3=5
→ Multiply by 2: x+6=10→x=4x+6=10→x=4.
