Linear Inequalities in One or Two Variables

Linear Inequalities in One or Two Variables

SAT Math | Algebra | Foundation to Advanced

Linear inequalities describe relationships in which one quantity is greater than, less than, greater than or equal to, or less than or equal to another quantity. On the SAT, you must be able to solve inequalities, understand their graphs, test possible solutions, translate word problems into inequalities, and interpret regions that satisfy one or more conditions.

1. Understanding Linear Inequalities

A linear inequality is similar to a linear equation, but it uses an inequality symbol instead of an equals sign.

An equation describes an equality.

For example:

3x + 5 = 20

An inequality describes a range of possible values.

For example:

3x + 5 > 20

This means that 3x + 5 must be greater than 20.

Unlike an equation with exactly one solution, an inequality often has infinitely many possible solutions.

The four inequality symbols

Consider:

x > 5

This means x can be any number greater than 5, but x cannot equal 5.

Examples of solutions include 6, 8, 10, and 5.5.

Now consider:

x ≥ 5

This means x can equal 5 or any number greater than 5.

Key idea: The symbols ≤ and ≥ include equality, while < and > do not.

SAT vocabulary

SAT word problems often use phrases that indicate an inequality.

Recognizing this vocabulary is important when creating mathematical models from real-world situations.

2. Solving Basic Linear Inequalities

Solving a linear inequality is similar to solving a linear equation.

The goal is to isolate the variable.

You may add or subtract the same number on both sides without changing the direction of the inequality symbol.

You may also multiply or divide both sides by the same positive number without changing the inequality symbol.

Worked Example 1 — Basic inequality

Difficulty: Easy

Solve:

3x + 7 > 19

Step 1: Subtract 7 from both sides.

3x > 12

Step 2: Divide by 3.

x > 4

Final answer: x > 4

This means every number greater than 4 is a solution.

For example:

x = 5 satisfies the inequality because:

3(5) + 7 = 22

22 > 19

However, x = 4 does not satisfy the strict inequality because:

3(4) + 7 = 19

19 is not greater than 19.

Worked Example 2 — Less than or equal to

Difficulty: Easy

Solve:

5x − 6 ≤ 24

Step 1: Add 6.

5x ≤ 30

Step 2: Divide by 5.

x ≤ 6

Final answer: x ≤ 6

The number 6 is included because the inequality contains ≤.

SAT tip: Pay attention to whether the boundary value is included. A solution such as x ≤ 6 is different from x < 6.

3. Multiplying or Dividing by Negative Numbers

This is one of the most important rules in linear inequalities.

Whenever you multiply or divide both sides of an inequality by a negative number, reverse the direction of the inequality symbol.

For example:

−2x > 8

Divide both sides by −2.

Because the divisor is negative, reverse > to <.

x < −4

Why does the inequality reverse?

Consider the true statement:

2 < 5

Multiply both sides by −1.

−2 > −5

The direction reverses because negative multiplication reverses the order of numbers.

Worked Example 3 — Negative coefficient

Difficulty: Medium

Solve:

5 − 3x ≥ 20

Step 1: Subtract 5.

−3x ≥ 15

Step 2: Divide by −3.

Reverse the inequality symbol.

x ≤ −5

Final answer: x ≤ −5

Check using x = −6:

5 − 3(−6) = 23

23 ≥ 20

The inequality is satisfied.

Important reminder

Adding or subtracting a negative number does not reverse the symbol.

Only multiplying or dividing both sides by a negative number reverses it.

4. Multi-Step Inequalities

Some SAT questions contain parentheses, variables on both sides, or several operations.

The general process is:

1. Expand parentheses when necessary.

2. Combine like terms.

3. Collect variable terms on one side.

4. Collect constants on the other side.

5. Isolate the variable.

6. Reverse the inequality symbol if you multiply or divide by a negative number.

Worked Example 4 — Variables on both sides

Difficulty: Medium

Solve:

4x − 7 ≤ 2x + 9

Step 1: Subtract 2x from both sides.

2x − 7 ≤ 9

Step 2: Add 7.

2x ≤ 16

Step 3: Divide by 2.

x ≤ 8

Final answer: x ≤ 8

Worked Example 5 — Parentheses and a negative coefficient

Difficulty: Hard

Solve:

4 − 3(2x − 1) ≤ 2x + 15

Step 1: Distribute −3.

4 − 6x + 3 ≤ 2x + 15

Step 2: Combine constants.

7 − 6x ≤ 2x + 15

Step 3: Subtract 2x.

7 − 8x ≤ 15

Step 4: Subtract 7.

−8x ≤ 8

Step 5: Divide by −8 and reverse the inequality symbol.

x ≥ −1

Final answer: x ≥ −1

Common mistake: Forgetting that the negative coefficient outside the parentheses must be distributed to every term.

5. Inequalities Containing Fractions and Decimals

Fractional and decimal inequalities follow the same mathematical rules as other inequalities.

Solving fractional inequalities

A useful strategy is to multiply every term by the least common denominator.

If the number used to clear denominators is positive, the inequality direction stays the same.

Worked Example 6 — Fractional inequality

Difficulty: Medium

Solve:

(3x − 2)/4 ≥ (x + 5)/2

Step 1: Identify the least common denominator.

LCD = 4

Step 2: Multiply both sides by 4.

3x − 2 ≥ 2(x + 5)

Step 3: Distribute.

3x − 2 ≥ 2x + 10

Step 4: Subtract 2x.

x − 2 ≥ 10

Step 5: Add 2.

x ≥ 12

Final answer: x ≥ 12

Worked Example 7 — Decimal inequality

Difficulty: Medium

Solve:

0.5x + 2.4 < 8.4

Step 1: Subtract 2.4.

0.5x < 6

Step 2: Divide by 0.5.

x < 12

Final answer: x < 12

Alternatively, multiply the original inequality by 10:

5x + 24 < 84

5x < 60

x < 12

Both approaches produce the same solution.

6. Graphing Inequalities on a Number Line

A one-variable inequality can be represented using a number line.

The graph shows all possible values that satisfy the inequality.

Open circle

An open circle means the boundary value is not included.

Use an open circle with:

< and >

For example:

x > 3

Place an open circle at 3 and shade to the right.

Closed circle

A closed circle means the boundary value is included.

Use a closed circle with:

≤ and ≥

For example:

x ≤ 3

Place a closed circle at 3 and shade to the left.

Direction of shading

For x > a or x ≥ a, shade to the right.

For x < a or x ≤ a, shade to the left.

Worked Example 8 — Interpreting a number line

Difficulty: Easy

A number line has a closed circle at −2 and shading extending to the right.

Which inequality does it represent?

Because the circle is closed, −2 is included.

Because the shading extends right, the values are greater than −2.

Final answer: x ≥ −2

SAT tip: A number-line graph communicates both the range of solutions and whether the boundary value is included.

7. Compound Linear Inequalities

A compound inequality contains two inequality conditions involving the same variable.

For example:

2 < x ≤ 7

This means:

x > 2

AND

x ≤ 7

Both conditions must be satisfied.

Examples of valid values include:

3, 4, 5, 6, and 7.

The value 2 is not included.

Worked Example 9 — Compound inequality

Difficulty: Medium

Solve:

−2 ≤ 3x + 4 < 10

Step 1: Subtract 4 from all three parts.

−6 ≤ 3x < 6

Step 2: Divide all three parts by 3.

−2 ≤ x < 2

Final answer: −2 ≤ x < 2

The solution includes −2 but excludes 2.

Special consideration

If you multiply or divide all parts of a compound inequality by a negative number, reverse both inequality symbols and rewrite the results in the correct order.

Compound inequalities are useful for representing ranges, acceptable limits, and intervals.

8. Linear Inequalities in Real-World Problems

Many SAT inequalities describe restrictions or requirements.

Common examples include:

• Budgets

• Maximum capacity

• Minimum scores

• Age restrictions

• Production limits

• Quantities of products

• Time limits

Worked Example 10 — Budget constraint

Difficulty: Medium

A student has $95 to spend on notebooks.

Each notebook costs $7, and there is a one-time delivery fee of $11.

What is the greatest number of notebooks the student can purchase?

Step 1: Let n represent the number of notebooks.

Step 2: Write the inequality.

7n + 11 ≤ 95

Step 3: Subtract 11.

7n ≤ 84

Step 4: Divide by 7.

n ≤ 12

Because the number of notebooks must be a whole number, the greatest possible quantity is 12.

Final answer: 12 notebooks

Worked Example 11 — Minimum requirement

Difficulty: Medium

A student needs at least 180 points to earn a reward.

The student already has 36 points and earns 12 additional points for each completed activity.

What is the minimum number of activities required?

Step 1: Let a represent the number of activities.

Step 2: Write the inequality.

36 + 12a ≥ 180

Step 3: Subtract 36.

12a ≥ 144

Step 4: Divide by 12.

a ≥ 12

Final answer: 12 activities

Interpreting whole-number answers

When a problem asks for the maximum or minimum number of whole items, interpret the algebraic solution carefully.

For example:

n ≤ 7.8

If n must be a nonnegative integer, the greatest possible value is 7.

If the result is:

n ≥ 7.2

the smallest possible integer value is 8.

9. Linear Inequalities in Two Variables

A linear inequality in two variables contains two variables, usually x and y.

Examples include:

y > 2x + 1

3x + 2y ≤ 12

x − y ≥ 5

A solution is an ordered pair (x, y) that makes the inequality true.

Unlike a linear equation, which represents a line, a linear inequality in two variables represents a region of the coordinate plane.

Worked Example 12 — Testing an ordered pair

Difficulty: Medium

Determine whether the point (2, 5) satisfies:

y > 2x − 1

Step 1: Substitute x = 2 and y = 5.

5 > 2(2) − 1

Step 2: Simplify.

5 > 3

This is true.

Conclusion: (2, 5) is a solution.

Checking another point

Consider (2, 3).

Substitute:

3 > 2(2) − 1

3 > 3

This is false.

Therefore, (2, 3) is not a solution.

Notice that (2, 3) lies on the boundary line, but the strict inequality does not include that line.

SAT tip: For questions asking which point satisfies an inequality, substituting each answer choice is often faster than graphing.

10. Graphing Linear Inequalities in Two Variables

To graph a linear inequality in two variables, begin by drawing its boundary line.

For example:

y > 2x + 1

The boundary line is:

y = 2x + 1

The inequality divides the coordinate plane into two regions.

One region satisfies the inequality.

The other does not.

Step 1 — Identify the boundary line

Replace the inequality sign with an equals sign.

Example:

y ≤ −x + 4

Boundary:

y = −x + 4

Step 2 — Determine the boundary style

Use a dashed line when the inequality is strict:

< or >

Use a solid line when equality is included:

≤ or ≥

Step 3 — Determine the shading

If the inequality is:

y > mx + b

shade above the line.

If the inequality is:

y < mx + b

shade below the line.

The same directions apply to ≥ and ≤, but the boundary is included.

Worked Example 13 — Graph interpretation

Difficulty: Medium

Describe the graph of:

y ≥ −x + 4

Step 1 — Find the boundary.

y = −x + 4

Step 2 — Determine the line style.

The inequality includes equality.

Therefore, the line is solid.

Step 3 — Determine the shaded region.

The inequality uses ≥.

Therefore, shade above the line.

Final answer: A solid boundary line y = −x + 4, with the region above it shaded.

Testing a point

The point (0, 0) gives:

0 ≥ −0 + 4

0 ≥ 4

This is false.

Therefore, the origin is not part of the solution region.

11. Rewriting Two-Variable Inequalities

Some inequalities are written in standard form rather than with y isolated.

For example:

3x − 2y ≥ 8

To understand the graph, rewrite the inequality in terms of y.

Worked Example 14 — Rearranging an inequality

Difficulty: Hard

Rewrite:

3x − 2y ≥ 8

in terms of y.

Step 1: Subtract 3x.

−2y ≥ 8 − 3x

Step 2: Divide by −2.

Because the divisor is negative, reverse the inequality.

y ≤ (3/2)x − 4

Final answer: y ≤ (3/2)x − 4

Interpreting the graph

The boundary line is:

y = (3/2)x − 4

The slope is 3/2.

The y-intercept is −4.

The boundary is solid because equality is included.

The solution region is below the line.

Common mistake: Dividing by a negative coefficient without reversing the inequality sign.

12. Systems of Linear Inequalities

A system of linear inequalities contains two or more inequalities that must be satisfied at the same time.

For example:

y ≥ 2x − 1

y < −x + 8

A solution must satisfy both conditions.

Graphically, the solutions lie in the overlapping shaded regions.

Worked Example 15 — Testing a point in a system

Difficulty: Hard

Does (1, 2) satisfy the system?

y ≥ 2x − 1

y < −x + 8

Step 1 — Check the first inequality.

2 ≥ 2(1) − 1

2 ≥ 1

This is true.

Step 2 — Check the second inequality.

2 < −1 + 8

2 < 7

This is also true.

Conclusion: (1, 2) satisfies the system.

Important rule

If a point fails even one inequality in a system, it is not a solution to the entire system.

Real-world example

A school purchases notebooks and folders.

Each notebook costs $4.

Each folder costs $6.

The school has a budget of at most $120.

Let:

n = number of notebooks

f = number of folders

The budget constraint is:

4n + 6f ≤ 120

Because physical quantities cannot be negative:

n ≥ 0

f ≥ 0

Together, these inequalities describe the possible purchase combinations.

For a point to represent a valid purchase, n and f must also be whole numbers.

13. Connecting Tables, Equations, and Graphs

The SAT may describe an inequality through a table of points or a graph instead of giving the inequality directly.

You must connect the different representations.

Worked Example 16 — Finding an inequality from a boundary

Difficulty: Hard

A dashed boundary line passes through the points:

(0, 5) and (2, 1)

The shaded region is below the line.

Which inequality represents the region?

Step 1 — Calculate the slope.

m = (1 − 5)/(2 − 0)

m = −4/2

m = −2

Step 2 — Identify the y-intercept.

The line passes through (0, 5).

Therefore, b = 5.

Step 3 — Write the boundary equation.

y = −2x + 5

Step 4 — Determine the inequality symbol.

The boundary is dashed, so equality is excluded.

The region is below the line, so use <.

Final answer: y < −2x + 5

Interpreting a table of possible solutions

If a table provides several ordered pairs, substitute each point into the inequality.

Check which points satisfy the relationship.

When a point lies exactly on the boundary, determine whether equality is included before accepting it.

14. Advanced SAT Strategies

Strategy 1 — Recognize inequality vocabulary

"At least" indicates ≥.

"At most" indicates ≤.

"More than" indicates >.

"Fewer than" indicates <.

Strategy 2 — Track negative coefficients

Reversing the symbol when dividing or multiplying by a negative number is essential.

Strategy 3 — Test answer choices directly

For a two-variable inequality, substituting points may be faster than drawing the entire graph.

Strategy 4 — Identify boundary inclusion

A solid line includes boundary points.

A dashed line excludes boundary points.

Strategy 5 — Consider integer restrictions

When counting objects, people, tickets, or completed activities, the answer often must be a nonnegative whole number.

Strategy 6 — Interpret systems carefully

A point must satisfy every inequality to belong to a system's solution region.

Strategy 7 — Use Desmos when helpful

The Digital SAT includes an embedded Desmos graphing calculator.

Desmos can help visualize inequalities and the overlap of shaded regions.

However, direct algebra or substitution may be faster for questions involving one variable or a small set of possible points.

15. Common SAT Mistakes

Mistake 1 — Forgetting to reverse the inequality

Incorrect:

−3x > 12 becomes x > −4

Correct:

−3x > 12 becomes x < −4

Mistake 2 — Confusing at least and at most

At least 20 means x ≥ 20.

At most 20 means x ≤ 20.

Mistake 3 — Including the boundary in a strict inequality

x > 5 does not include x = 5.

y < 2x + 1 does not include points exactly on the boundary line.

Mistake 4 — Choosing the wrong shading direction

For y > mx + b, shade above.

For y < mx + b, shade below.

Mistake 5 — Forgetting to distribute a negative number

Incorrect:

−2(x − 4) = −2x − 8

Correct:

−2(x − 4) = −2x + 8

Mistake 6 — Reversing x and y when testing points

For (3, 5):

x = 3

y = 5

Do not substitute them in the opposite order.

Mistake 7 — Checking only one inequality in a system

A valid point must satisfy every condition.

Mistake 8 — Ignoring real-world restrictions

A mathematical result of 7.5 notebooks is not a valid whole-item purchase.

Mistake 9 — Confusing solid and dashed lines

Solid means equality is included.

Dashed means equality is excluded.

Mistake 10 — Assuming a boundary line is the entire solution

A two-variable linear inequality represents a region of the plane, not only the boundary line.

16. Check Your Understanding

Question 1 — Negative coefficient

Solve:

−4x + 5 > 21

A. x > −4

B. x < −4

C. x ≤ −4

D. x ≥ −4

Correct answer: B

Subtract 5:

−4x > 16

Divide by −4 and reverse the symbol:

x < −4

Question 2 — Testing an ordered pair

Which point satisfies:

y ≤ 2x − 1?

A. (0, 0)

B. (1, 2)

C. (2, 3)

D. (−1, 0)

Correct answer: C

For (2, 3):

3 ≤ 2(2) − 1

3 ≤ 3

This is true.

Question 3 — Graphing

A graph has a dashed line:

y = −x + 3

and the region above the line is shaded.

What is the inequality?

Correct answer: y > −x + 3

The dashed line means the boundary is excluded.

The shading above the line indicates >.

Question 4 — Maximum integer

What is the greatest integer x satisfying:

4x + 6 ≤ 38?

Correct answer: 8

Subtract 6:

4x ≤ 32

Divide by 4:

x ≤ 8

The greatest integer solution is 8.

Question 5 — System of inequalities

Does (3, 4) satisfy both inequalities?

x ≥ 0

2x + 3y ≤ 18

Correct answer: Yes

The first inequality is satisfied because 3 ≥ 0.

For the second:

2(3) + 3(4) = 6 + 12 = 18

18 ≤ 18 is true.

Therefore, (3, 4) satisfies the system.

17. Lesson Summary

Linear inequalities compare quantities using <, >, ≤, and ≥.

A linear inequality in one variable usually represents a range of possible values.

Adding or subtracting the same number from both sides preserves the inequality direction.

Multiplying or dividing by a negative number reverses the inequality symbol.

Strict inequalities exclude their boundary values.

Inclusive inequalities contain their boundary values.

A two-variable inequality represents a shaded region in the coordinate plane.

Dashed boundary lines exclude equality.

Solid boundary lines include equality.

An ordered pair satisfies an inequality when substitution produces a true statement.

A solution to a system of inequalities must satisfy every inequality in that system.

SAT word problems frequently use inequalities to model budgets, minimum requirements, maximum capacities, and other constraints.

You have now completed the five core SAT Algebra topics and are ready to practice Linear Inequalities in One or Two Variables in Baloo Prep.