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Linear
Inequalities

Master the rules for solving and graphing linear inequalities: a core SAT Math skill tested in every exam.

SAT Math

What Is a Linear Inequality?

A linear inequality looks just like a linear equation, but instead of \( = \), it uses one of four inequality symbols: \( < \), \( > \), \( \leq \), or \( \geq \).

\( < \) and \( > \)
Strict inequalities: the boundary value is not included.
\( \leq \) and \( \geq \)
Non-strict inequalities: the boundary value is included.

Visualizing on a Number Line

The solution to a linear inequality is a range of values, not a single point. We show this range on a number line.

0 2 4 6 8 \( x < 4 \) Open circle = not included 0 2 4 6 8 \( x \geq 3 \) Filled circle = included

Why This Matters on the SAT

Linear inequalities appear in multiple SAT question types. Understanding them unlocks points across the entire math section.

Algebraic
Solve for \( x \) in an inequality and identify the solution set.
Graphical
Shade the correct region on a coordinate plane or number line.
Word Problems
Translate real-world constraints into inequality expressions.

Core Rules for Solving

You solve a linear inequality exactly like an equation, with one critical exception.

1
Add or subtract the same value on both sides: the inequality direction stays the same.
2
Multiply or divide by a positive number: the inequality direction stays the same.
3
Multiply or divide by a negative number: you must flip the inequality sign.

The Flip Rule Visualized

Solving: \( -3x + 6 \leq 15 \) Step 1: Subtract 6 from both sides \( -3x \leq 9 \) Step 2: Divide both sides by -3 \( x \geq -3 \) FLIP -6 -4 -2 0 2 \( x \geq -3 \) Filled circle because \( \leq \) became \( \geq \) (boundary included)

Equations vs. Inequalities

Equation Inequality
Symbol \( = \) \( <, >, \leq, \geq \)
Solution One value (usually) A range of values
Graph A single point A ray or segment
Multiply by negative No change Flip the sign
Notation \( x = 5 \) \( x > 5 \) or \( (5, \infty) \)

How to Solve Any Linear Inequality

1
Simplify each side: distribute and combine like terms.
2
Isolate the variable term: add or subtract to move constants to the other side.
3
Divide (or multiply) to solve: if the coefficient is negative, flip the sign.
4
Check with a test value: pick a number in your solution range and plug it back in.
5
Graph or express the answer: use a number line, interval notation, or set notation.

Example 1: Basic Inequality

1
Add 7 to both sides: \( 4x > 16 \)
2
Divide both sides by 4 (positive, no flip): \( x > 4 \)
3
Check: try \( x = 5 \): \( 4(5) - 7 = 13 > 9 \) ✓
0 2 4 6 8 \( x > 4 \)

Example 2: Flipping the Sign

1
Subtract 2 from both sides: \( -5x \geq 15 \)
2
Divide both sides by \( -5 \) and flip: \( x \leq -3 \)
3
Check: try \( x = -4 \): \( -5(-4) + 2 = 22 \geq 17 \) ✓
-6 -4 -2 0 2 \( x \leq -3 \)

Example 3: Multi-Step Inequality

1
Distribute: \( 6x - 3 < 4x + 9 \)
2
Subtract \( 4x \) from both sides: \( 2x - 3 < 9 \)
3
Add 3 to both sides: \( 2x < 12 \)
4
Divide by 2 (positive, no flip): \( x < 6 \)
5
Check: try \( x = 0 \): \( 3(2 \cdot 0 - 1) = -3 < 4(0) + 9 = 9 \) ✓

Graphing on the Coordinate Plane

A two-variable inequality like \( y < 2x + 1 \) defines a region on the coordinate plane. The boundary line is dashed (strict) or solid (non-strict), and the solution is the shaded side.

x y 0 1 2 3 1 2 3 -1 -1 \( y = 2x + 1 \) \( y < 2x + 1 \) (shaded region) Dashed line: boundary not included ( \( < \) )

Compound Inequalities

A compound inequality combines two inequalities. On the SAT, the most common form is the "sandwich" type: \( a < x < b \).

AND (Intersection)
\( -2 < x \leq 5 \): values that satisfy both conditions. Graph is a segment.
OR (Union)
\( x < -1 \) or \( x > 3 \): values that satisfy either condition. Graph is two rays.
-4 -2 0 2 4 6 \( -2 < x \leq 5 \) (AND) -4 -2 0 2 4 6 \( x < -1 \) or \( x > 3 \) (OR)

Translating Word Problems

1
Define the variable: Let \( t \) = number of texts.
2
Write the inequality: \( 25 + 0.10t \leq 50 \)
3
Solve: \( 0.10t \leq 25 \) then \( t \leq 250 \)
4
Interpret: You can send at most 250 texts.

Common Mistakes to Avoid

✕ Forgetting to Flip
When dividing by a negative, the sign must reverse. This is the single most common error on SAT inequality questions.
✕ Open vs. Filled Circle
Using a filled circle for \( < \) or \( > \), or an open circle for \( \leq \) or \( \geq \). Match the symbol to the circle type.
✕ Wrong Shading Direction
Shading above the line for \( y < mx + b \) instead of below it. Always test the origin \( (0,0) \) to check which side to shade.

Quick Reference Summary

Symbol Meaning Circle Line Type
\( < \) Less than Open Dashed
\( > \) Greater than Open Dashed
\( \leq \) Less than or equal to Filled Solid
\( \geq \) Greater than or equal to Filled Solid

Question 1 of 5

What is the solution to \( 3x - 8 > 7 \)?

Question 2 of 5

Solve for \( x \): \( -2x + 6 \leq 12 \)

Question 3 of 5

Which phrase translates to \( x \leq 12 \)?

Question 4 of 5

If the graph of a linear inequality uses a dashed line, which symbols could the inequality use?

Question 5 of 5

A gym membership costs \( \$30 \) per month plus \( \$5 \) per class. You want to spend less than \( \$80 \) per month. Which inequality models the number of classes \( c \) you can take?