Inequalities show when one quantity is greater than, less than, or not equal to another quantity. This cheat sheet helps students solve, graph, and write inequalities using symbols, number lines, and interval notation. It is especially useful when checking when an answer should include an endpoint or extend forever.
Students in grades 8 through 10 use these skills in algebra, graphing, word problems, and functions.
The most important rule is that multiplying or dividing both sides by a negative number reverses the inequality sign. Compound inequalities use the words and and or to describe overlapping or combined solution sets. Absolute value means distance from , so usually creates two possible equations when .
Absolute value inequalities follow two main patterns: means , while means or .
Key Facts
- The symbols , , , and mean less than, greater than, less than or equal to, and greater than or equal to.
- When adding or subtracting the same value on both sides, the inequality direction stays the same, such as .
- When multiplying or dividing both sides by a negative number, reverse the inequality sign, such as .
- Use an open circle for or and a closed circle for or on a number line.
- A compound and inequality means both conditions must be true, such as .
- A compound or inequality means at least one condition must be true, such as or .
- For , the equation has two solutions: or .
- For , becomes , and becomes or .
Vocabulary
- Inequality
- An inequality is a mathematical statement that compares two expressions using symbols such as , , , or .
- Solution set
- A solution set is the collection of all values that make an equation or inequality true.
- Interval notation
- Interval notation describes a set of numbers using parentheses for excluded endpoints and brackets for included endpoints, such as .
- Compound inequality
- A compound inequality combines two inequalities using and or or, such as and .
- Absolute value
- Absolute value is the distance of a number from on a number line, written as .
- Endpoint
- An endpoint is a boundary value of an interval, and it is included when the symbol is or .
Common Mistakes to Avoid
- Not flipping the inequality when multiplying or dividing by a negative; this is wrong because becomes , not .
- Using a closed circle for or ; this is wrong because strict inequalities do not include the endpoint.
- Solving with only one answer; this is wrong because when , gives or .
- Writing as or ; this is wrong because less than means the solution is between the two boundary values, so .
- Treating and and or the same way; this is wrong because and means overlap, while or means combine all values from either inequality.
Practice Questions
- 1 Solve and graph the inequality .
- 2 Solve the compound inequality and write the answer in interval notation.
- 3 Solve the absolute value inequality .
- 4 Explain why multiplying both sides of by changes the inequality direction.
Understanding Inequalities & Absolute Value
An inequality is best understood as a rule for a whole set of numbers, not as a problem with one final number. Solving it means finding every value that makes the statement true. A quick check helps catch mistakes.
Pick one number from your proposed solution region and substitute it into the original statement. Then try a number outside the region.
The first should work and the second should fail. This is especially important after several algebra steps, because one small sign error can change an entire half of the number line.
The reversal rule for negative numbers comes from order. On a number line, multiplying by a positive value keeps numbers in the same left to right order. Multiplying by a negative value reflects them across zero.
For example, negative three is less than negative one. After multiplying both values by negative two, six is greater than two. The order has flipped.
Remembering this number line movement is safer than treating the rule as a fact to memorize. It applies only when every part of the inequality is multiplied or divided by a negative value.
Interval notation is a compact way to record the same answer shown on a number line. Parentheses mean an endpoint is not included. Square brackets mean it is included.
Infinity always uses a parenthesis because infinity is not a reachable number or an endpoint that can be included. For two separate pieces, use the union symbol, which means combine the sets. Students often make errors by writing the endpoints correctly but choosing the wrong bracket.
Connect each bracket choice to the original wording. Strict limits exclude a boundary. Limits that allow equality include it.
Absolute value becomes useful when the important idea is distance rather than direction. A temperature can be a certain number of degrees away from zero. A measurement can be within a chosen error tolerance.
A speed can differ from a target by no more than a certain amount. In these cases, the expression inside absolute value describes a difference, while the outside value tells how far away it is. An equation with absolute value can produce two locations equally far from a center.
An inequality can describe values close to the center or values far from it. First isolate the absolute value expression when possible.
Then decide whether the situation describes staying within a distance or being beyond a distance. Sketching the center and boundary points on a number line makes the correct region much easier to see.