GED Math: Inequalities
Real-life limits lead to inequalities: at most or no more than (a maximum) uses ≤, and at least or no fewer than (a minimum) uses ≥. Define the variable, write the expression for the quantity that is limited, and compare it with the limit. Solve as usual and then interpret the answer in context: if the variable counts people or items, only whole numbers make sense, so you may round the boundary down for a maximum or up for a minimum.
1. Inequality Symbols and What They Mean
An inequality compares two expressions that are not necessarily equal. The four symbols are: < less than, > greater than, ≤ less than or equal to, and ≥ greater than or equal to. The small end of the symbol always points to the smaller quantity, so 3 < 8 and 8 > 3 say the same thing. Translate key phrases carefully: at least means ≥; at most and no more than mean ≤; more than and greater than mean >; fewer than and less than mean <. Remember that on the number line, numbers farther to the left are smaller, so −7 < −2.
Worked Example: Write “the sum of a number and 5 is at most 20.”
“The sum of a number and 5” is n + 5, and “at most” means less than or equal to. The inequality is n + 5 ≤ 20.
1. Which symbol means “greater than or equal to”?
Explanation: The bar under the > means “or equal to,” so ≥ is greater than or equal to.
2. Which statement is true?
Explanation: On the number line, −3 is to the left of −1, so −3 is smaller: −3 < −1.
3. Which inequality says “x is at least 12”?
Explanation: At least means 12 or more, including 12: x ≥ 12.
4. Which inequality says “a number n is no more than 9”?
Explanation: No more than means the number can be 9 or smaller: n ≤ 9.
5. Which inequality says “twice a number is more than 14”?
Explanation: Twice a number is 2n, and “more than 14” means > 14 (14 is not included).
6. Which inequality is the same as 8 > x?
Explanation: Reading it the other way: 8 is greater than x means x is less than 8.
7. An elevator can hold at most 12 people. If p is the number of people, which inequality applies?
Explanation: At most 12 means 12 or fewer: p ≤ 12.
2. Solving One- and Multi-Step Inequalities
Solve an inequality just like an equation: use inverse operations to isolate the variable, doing the same thing to both sides. Distribute and combine like terms first, move variable terms to one side, then undo addition/subtraction and multiplication/division. The answer is usually a whole range of numbers, such as x < 2, meaning every number less than 2 works. To check, substitute a number from your solution set into the original inequality and make sure the statement is true. (Also test the boundary number: it should make the statement false for < or > and true for ≤ or ≥.)
Worked Example: Solve 3x + 2 < 8.
Subtract 2 from both sides: 3x < 6. Divide both sides by 3: x < 2. Check with x = 1: 3(1) + 2 = 5, and 5 < 8 is true ✓.
1. Solve 3x − 7 > 5
Explanation: Add 7: 3x > 12. Divide by 3: x > 4.
2. Solve 4x + 3 < 19
Explanation: Subtract 3: 4x < 16. Divide by 4: x < 4.
3. Solve 13 < 2x − 1
Explanation: Add 1: 14 < 2x. Divide by 2: 7 < x, which is the same as x > 7.
4. Solve 4 + x5 ≤ 8
Explanation: Multiply both sides by 5: 4 + x ≤ 40. Subtract 4: x ≤ 36.
5. Solve 2(x + 3) < 4
Explanation: Distribute: 2x + 6 < 4. Subtract 6: 2x < −2. Divide by 2: x < −1.
6. Solve 3 + 9x ≥ 4(x + 7)
Explanation: Distribute: 3 + 9x ≥ 4x + 28. Subtract 4x: 3 + 5x ≥ 28. Subtract 3: 5x ≥ 25. So x ≥ 5.
7. Solve 5x + 14 > 2 + 7x
Explanation: Subtract 5x: 14 > 2 + 2x. Subtract 2: 12 > 2x. So 6 > x, or x < 6.
8. Solve x + 6 ≤ 8x − 15
Explanation: Subtract x: 6 ≤ 7x − 15. Add 15: 21 ≤ 7x. So 3 ≤ x, or x ≥ 3.
3. Multiplying or Dividing by a Negative Number
There is one big difference between equations and inequalities: whenever you multiply or divide both sides by a negative number, you must reverse the inequality symbol. For example, 2 < 5 is true, but multiplying both sides by −1 gives −2 and −5, and −2 > −5, so the symbol flips. Adding or subtracting a negative number does not reverse the symbol. A common trick to avoid the flip is to move the variable term to the side where it ends up positive. Always check with a test value to catch an accidental flip (or a missed one).
Worked Example: Solve −2n − 5 ≥ 3.
Add 5: −2n ≥ 8. Divide both sides by −2 and reverse the symbol: n ≤ −4. Check with n = −5: −2(−5) − 5 = 5, and 5 ≥ 3 is true ✓.
1. Solve −2n − 5 ≥ 3
Explanation: Add 5: −2n ≥ 8. Divide by −2 and flip: n ≤ −4.
2. Solve −3n + 4 < 19
Explanation: Subtract 4: −3n < 15. Divide by −3 and flip: n > −5.
3. Solve −2x + 9 < 1
Explanation: Subtract 9: −2x < −8. Divide by −2 and flip: x > 4.
4. Solve −4(x + 2) < 24
Explanation: Divide both sides by −4 and flip: x + 2 > −6. Subtract 2: x > −8.
5. Solve 36 > 4(x − 12)
Explanation: Divide by 4 (positive, no flip): 9 > x − 12. Add 12: 21 > x, or x < 21.
6. Solve −x3 ≥ 4
Explanation: Multiply both sides by 3: −x ≥ 12. Multiply by −1 and flip: x ≤ −12.
7. Solve −5 + 3x ≥ 4(3x − 8)
Explanation: Distribute: −5 + 3x ≥ 12x − 32. Subtract 3x and add 32: 27 ≥ 9x. So x ≤ 3 (dividing by positive 9, no flip needed).
8. When must you reverse the direction of the inequality symbol?
Explanation: Only multiplying or dividing by a negative flips the symbol. Adding and subtracting do not.
9. Which number is a solution of −2x + 9 < 1?
Explanation: Try x = 5: −10 + 9 = −1 < 1 is true. For x = 4: −8 + 9 = 1 < 1 is false.
4. Graphing Inequalities on a Number Line
Because an inequality usually has many solutions, you can show them on a number line. Draw a circle at the boundary number: an open circle means the number is not included (< or >), and a closed (filled) circle means it is included (≤ or ≥). Then shade or draw an arrow in the direction of the solutions: to the left for < and ≤, to the right for > and ≥. Read the graph in the reverse direction too: an arrow pointing right means “greater,” and a filled dot means “or equal to.”
Worked Example: Graph the solution of 2x + 1 ≥ −5.
Subtract 1: 2x ≥ −6. Divide by 2: x ≥ −3. Draw a closed circle at −3 and shade to the right.
The four basic graphs look like this:
1. Which inequality does the graph show?
Explanation: An open circle at 2 (not included) with the arrow to the left means x < 2.
2. Which inequality does the graph show?
Explanation: An open circle at −3 with the arrow to the right means x > −3.
3. Which inequality does the graph show?
Explanation: A closed circle at 1 with the arrow to the left means x ≤ 1.
4. Which inequality does the graph show?
Explanation: A closed circle at −2 with the arrow to the right means x ≥ −2.
5. Which graph description matches the solution of 2x − 3 < 5?
Explanation: Add 3: 2x < 8. Divide by 2: x < 4. Less than means an open circle at 4 and shading left.
6. Which graph description matches the solution of −3x ≤ 6?
Explanation: Divide by −3 and flip: x ≥ −2. Closed circle at −2, shaded right.
5. Compound Inequalities
A compound inequality combines two inequalities, such as 2 < x < 4, which means x is greater than 2 and less than 4 at the same time. To solve one written in three parts, do the same operation to all three parts (and flip both symbols if you multiply or divide by a negative). To solve one written as two separate inequalities, solve each and combine the results. On a number line, the solution is a segment between two circles (open for <, closed for ≤).
Worked Example: Solve 3x + 4 < 5x < 16 + x.
Split it: 3x + 4 < 5x gives 4 < 2x, so 2 < x. And 5x < 16 + x gives 4x < 16, so x < 4. Together: 2 < x < 4.
1. Solve 3x + 4 < 5x < 16 + x
Explanation: Solve each part: 3x + 4 < 5x gives x > 2, and 5x < 16 + x gives x < 4.
2. Solve −3 ≤ 2x + 1 < 9
Explanation: Subtract 1 from all parts: −4 ≤ 2x < 8. Divide all parts by 2: −2 ≤ x < 4.
3. Solve 5 < 3x − 4 ≤ 17
Explanation: Add 4 to all parts: 9 < 3x ≤ 21. Divide by 3: 3 < x ≤ 7.
4. How many integers satisfy −2 < x ≤ 3?
Explanation: The integers are −1, 0, 1, 2, 3 (not −2, because < excludes it). That is 5.
5. Which number makes 2 < x < 4 true?
Explanation: 3.5 is greater than 2 and less than 4. The endpoints 2 and 4 are excluded.
6. Which compound inequality does the graph show?
Explanation: Open circle at −1 (not included) and closed circle at 3 (included): −1 < x ≤ 3.
6. Inequality Word Problems
Real-life limits lead to inequalities: at most or no more than (a maximum) uses ≤, and at least or no fewer than (a minimum) uses ≥. Define the variable, write the expression for the quantity that is limited, and compare it with the limit. Solve as usual and then interpret the answer in context: if the variable counts people or items, only whole numbers make sense, so you may round the boundary down for a maximum or up for a minimum.
Worked Example: The perimeter of a square is at most 64 inches. What are the possible side lengths s?
Perimeter = 4s, so 4s ≤ 64. Divide by 4: s ≤ 16. The side can be 16 inches or shorter (and greater than 0).
1. An equilateral triangle has a perimeter of at most 45 inches. If s is the side length, which inequality gives the possible values of s?
Explanation: The perimeter is 3s, so 3s ≤ 45. Divide by 3: s ≤ 15.
2. Dana has $60. A movie ticket costs $12 and snacks cost $4 each. What is the greatest number of snacks n she can buy?
Explanation: 12 + 4n ≤ 60, so 4n ≤ 48 and n ≤ 12.
3. A phone plan costs $20 per month plus $0.10 per text. Chris wants to spend at most $35 per month. What is the greatest number of texts t he can send?
Explanation: 20 + 0.10t ≤ 35, so 0.10t ≤ 15 and t ≤ 150.
4. Mia scored 72, 85, and 78 on three tests. What is the lowest score she can earn on the fourth test and still have an average of at least 80?
Explanation: (72 + 85 + 78 + x) ÷ 4 ≥ 80 means 235 + x ≥ 320, so x ≥ 85.
5. A club sells shirts for $8 each and wants to raise at least $200. What is the least number of shirts n it must sell?
Explanation: 8n ≥ 200, so n ≥ 25.
6. Three added to the product of 4 and a number is less than 5 added to the product of 2 and the number. Which inequality is the solution for the number x?
Explanation: 3 + 4x < 5 + 2x gives 2x < 2, so x < 1.
Common Mistakes to Avoid
- Forgetting to reverse the symbol when multiplying or dividing both sides by a negative number.
- Reversing the symbol when you only add or subtract a negative number. That never flips the symbol.
- Reading the symbol backward: 8 > x means x is less than 8.
- Using an open circle for ≤ or ≥, or a closed circle for < or >.
- Shading the wrong direction on the number line. Check with a test point.
- Mistaking “at most” for ≥ and “at least” for ≤.
- In a three-part inequality, applying an operation to only the middle part.
- Forgetting to interpret the answer: counts of people or items must be whole numbers.
Frequently Asked Questions
What is an inequality?
A statement that compares two expressions using <, >, ≤, or ≥. It usually has many solutions.
When do I flip the inequality sign?
Only when you multiply or divide both sides by a negative number.
What do open and closed circles mean?
An open circle means the boundary number is not included (< or >). A closed circle means it is included (≤ or ≥).
How do I check my answer?
Pick a number from your solution set and substitute it into the original inequality. The statement should be true.
How do I solve a compound inequality?
Do the same operation to all three parts of a three-part inequality, or solve the two inequalities separately and combine the results.
What do “at least” and “at most” mean?
At least means greater than or equal to (≥). At most means less than or equal to (≤).
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.