Linear Inequalities

A linear inequality is what you get by replacing the equals sign in a linear equation with , , , or : , . Its solution is not a single number but a range of numbers — usually everything on one side of a boundary value.

The good news is that almost every technique from Linear Equations transfers unchanged. There is exactly one new rule, and it is the thing to get right: multiplying or dividing both sides by a negative number flips the inequality symbol.

The Linear Inequality Formula

The steps mirror solving . The only difference is the direction of the symbol in the answer, which depends on the sign of — because dividing by a negative reverses the inequality.

The One New Rule

Adding or subtracting any number leaves the symbol alone:

Multiplying or dividing by a positive number leaves the symbol alone:

Multiplying or dividing by a negative number reverses the symbol:

Why: on the number line, is true, but multiplying both sides by gives and , and now . Multiplying by a negative reflects every number across zero, which reverses their order. The inequality symbol has to reverse with it, or the statement becomes false. Test it on any pair: , multiply by , and .

The flip is tied to the operation you perform on the whole inequality, not to whether a negative sign appears somewhere. has a negative-looking , but you solve it by adding — no flip, so . requires multiplying both sides by — flip, so .

Graphing the Solution

Every solution is a ray on the number line: everything on one side of a single boundary value.

• or : open circle at the boundary (the boundary itself is not a solution), shade the ray toward the solutions.
• or : filled circle at the boundary (the boundary is included), shade toward the solutions.
• shades to the right (toward larger numbers); shades to the left.

The graph and the interval notation carry the same information: an open circle is a parenthesis, a filled circle is a bracket, and the shaded direction is the unbounded side of the interval.

Solving an Inequality with Fractions

Clear the denominators exactly as you would in an equation — multiply every term by the LCD of all the fractions. Because the LCD is a positive number, the symbol does not flip at that step.

The only place the flip rule can come up in a fraction problem is if a negative coefficient appears after clearing, and you have to divide by it at the end.

Interval Notation

A parenthesis means “not included”; a bracket means “included.” Infinity always takes a parenthesis. Full detail is in Interval Notation.

Worked Example A: A Positive Coefficient

Solve and give the answer in interval notation.

Step 1 — add to both sides:

Step 2 — divide by (positive — symbol stays):

Answer: , which is . Filled circle at , shaded left.

Worked Example B: Dividing by a Negative

Solve .

Step 1 — subtract :

Step 2 — divide by — the symbol flips:

Answer: , or . Open circle at , shaded left.

Worked Example C: Variables on Both Sides

Solve .

Step 1 — move the variable terms left, constants right:

Step 2 — divide by — the symbol flips:

Answer: , or .

Alternatively, moving the variable terms to the right (, then divide by positive ) gives , the same answer with no flip needed — moving variables to the side that keeps the coefficient positive avoids the flip entirely.

Worked Example D: All Reals or No Solution

Solve .

Step 1 — distribute and simplify:

The variable cancelled and left a true statement. Every real number satisfies the inequality.

Answer: all real numbers, . (Had it left , a false statement, the answer would be no solution, .)

Worked Example E: Clearing Fractions

Solve and give the answer in interval notation.

Step 1 — multiply every term by the LCD, (positive, so no flip):

Step 2 — distribute:

Step 3 — gather variable terms left, constants right:

Step 4 — divide by (positive, no flip):

Answer: , which is .

Worked Example F: Translating a Real-World Condition

A phone plan costs \$30 plus \$0.10 per text. For what number of texts does the monthly bill stay under \$45?

Step 1 — translate “bill stays under \$45”:

Step 2 — subtract :

Step 3 — divide by (positive, no flip):

Answer: fewer than texts, or if you note that cannot be negative. Word problems often add a natural lower bound like that the algebra alone would not produce.

Common Mistakes to Avoid

• Forgetting to flip when dividing by a negative. gives , not .
• Flipping when adding or subtracting a negative. gives ; no flip, because you added , you did not multiply by a negative.
• Flipping twice by mistake. Only the multiply/divide-by-negative step flips it. Doing it once per such step.
• Wrong bracket in interval notation. and get brackets; and get parentheses; infinity always a parenthesis.
• Shading the wrong way. shades right (toward larger numbers); shades left.
• Treating “all reals” or “no solution” as an error. Both are complete, correct answers.

Where This Shows Up Later

• Compound Inequalities. Two linear inequalities joined by “and” or “or”, each solved with this method.
• Absolute Value Inequalities. becomes the compound inequality .
• Polynomial and Rational Inequalities. These reduce to sign analysis, but each interval’s test is a linear inequality check.
• Linear programming. Systems of linear inequalities define feasible regions in two variables.
• Domain restrictions. “The radicand must be ” is a linear inequality you solve to find a function’s domain.

Practice Problems

Work each problem yourself before opening the answer. Give solutions in interval notation.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Problem 13. Solve .

Problem 14. A parking garage charges \$4 plus \$2 per hour. For how many hours is the total at most \$20?

Quick Reference

The solving steps are Linear Equations; the answer format is Interval Notation. Two of these joined together make a Compound Inequality. More Algebra lessons are available too.