Navigator
Math 141: Week 1
21 Sep 2026
1. **Inequality** — An inequality compares two quantities rather than saying they are exactly equal. The symbols are `<` (less than), `>` (greater than), `≤` (less than or equal to), and `≥` (greater than or equal to). A number is a **solution** if substituting it for the variable makes the inequality true. All numbers that work form the **solution set**. Different inequalities with exactly the same solution set are **equivalent inequalities**. 2. **Interval notation** — Interval notation is a compact way of writing sets of real numbers. Parentheses mean an endpoint is **not included**; brackets mean an endpoint **is included**. For example, `(a,b)` means `a < x < b`, while `[a,b]` means `a ≤ x ≤ b`. Mixed forms such as `[a,b)` and `(a,b]` include only one endpoint. 3. **Infinite intervals** — When an interval continues forever, infinity is written as `∞` or `−∞`. Examples are `(a,∞)`, `[a,∞)`, `(−∞,b)`, and `(−∞,b]`. Infinity is not an actual number or endpoint, so it **always gets a parenthesis**, never a bracket. `(−∞,∞)` represents all real numbers. 4. **Three ways to represent a solution set** — The same set can be shown with **inequality/set-builder notation**, **interval notation**, or a **number-line graph**. For example, `−1 < x ≤ 4` can be written `(-1,4]`. On the number line, an excluded endpoint is shown as open and an included endpoint as closed. 5. **Intersection of intervals** — The intersection, written `A ∩ B`, contains only the numbers that belong to **both** sets. Graphically, it is the overlapping part of the two number-line graphs. For example, the overlap of `(1,4)` and `[2,8]` is `[2,4)`. 6. **Union of intervals** — The union, written `A ∪ B`, contains every number that belongs to **either set or both**. Graphically, combine all portions covered by the two graphs. Overlapping intervals can sometimes combine into one larger interval. 7. **Linear inequality in one variable** — A linear inequality has a variable only to the first power and can be put into a form such as `ax + b < 0`, `ax + b ≤ 0`, `ax + b > 0`, or `ax + b ≥ 0`, with `a ≠ 0`. Solving it means isolating the variable while creating equivalent inequalities. 8. **Translating English into inequalities** — Common phrases correspond to inequality symbols. “At least” and “no less than” mean `≥`; “at most” and “no more than” mean `≤`. A phrase such as “between 5 and 7” can produce a compound inequality such as `5 < x < 7`. 9. **Addition property of inequalities** — You may add or subtract the same quantity on both sides of an inequality without changing its direction. This lets you move constant or variable terms while preserving the same solution set. 10. **Multiplying or dividing by a positive number** — Multiplying or dividing both sides by the same **positive** number preserves the direction of the inequality. If `a < b` and `c > 0`, then `ac < bc` and `a/c < b/c`. 11. **Multiplying or dividing by a negative number reverses the inequality** — This is the major difference between solving equations and solving inequalities. If both sides are multiplied or divided by a negative quantity, `<` becomes `>`, `≤` becomes `≥`, and vice versa. For example, dividing `−2x ≤ 8` by `−2` gives `x ≥ −4`. The reversal occurs because multiplication by a negative reflects numbers across zero on the number line. 12. **General procedure for solving a linear inequality** — The section's method is: simplify each side; collect variable terms on one side and constants on the other; isolate the variable; then write the answer in inequality/set-builder or interval notation and graph it. The variable does **not** have to end up on the left—the important thing is correctly interpreting the final inequality. 13. **Checking inequalities numerically and graphically** — A solution can be checked by substituting sample values into the original inequality. It can also be checked by graphing the two expressions: the solution consists of x-values where the graph representing one side lies above or below the other as required by the inequality. 14. **Inequalities containing fractions** — When constants appear in denominators, the fractions can often be cleared first by multiplying **every term on both sides** by the least common denominator. If that multiplier is positive, the inequality direction stays unchanged. After the fractions are removed, solve it like an ordinary linear inequality. 15. **No-solution inequalities** — Sometimes the variable disappears during simplification and leaves a false statement, such as `0 > 1`. Since that statement can never be true, **no real number satisfies the original inequality**. The solution set is the empty set, `∅`. 16. **Inequalities true for all real numbers** — If eliminating the variable produces a statement that is always true, such as `0 < 1`, then **every real number** satisfies the inequality. The answer can be written `ℝ` or `(−∞,∞)`. Graphically, this corresponds to the entire number line. 17. **Compound inequalities** — A compound inequality combines two comparisons, such as `−3 < 2x + 1 ≤ 3`. Although the word **and** is not written, it is implied: both conditions must be true simultaneously. To solve it, perform the same operation on **all three parts**, keeping the variable in the middle when convenient. 18. **Absolute value as distance** — `|x|` represents the distance between `x` and zero on the real number line. Because distance is nonnegative, absolute-value inequalities can be understood geometrically. For example, `|x| < 2` means x is less than 2 units from zero, so `−2 < x < 2`. 19. **Absolute-value inequalities using “and”** — For a positive number `c`, an inequality of the form `|u| < c` means the expression `u` lies **between** `−c` and `c`: `|u| < c ⇔ −c < u < c` The same idea works with `≤`: `|u| ≤ c` means `−c ≤ u ≤ c`. This normally produces one continuous interval. 20. **Absolute-value inequalities using “or”** — For positive `c`, `|u| > c` means the distance from zero is greater than `c`. Therefore the expression must lie on one side **or the other**: `|u| > c ⇔ u < −c or u > c` Likewise, `|u| ≥ c` produces `u ≤ −c or u ≥ c`. These usually produce two separate intervals joined by `∪`. 21. **Isolating the absolute-value expression first** — If an absolute-value inequality has coefficients or constants outside the bars, first isolate the absolute-value expression. Only then rewrite it as an ordinary compound inequality. If isolating it requires division by a negative number, the inequality sign must reverse before the absolute-value rule is applied. 22. **Graphical interpretation of absolute-value inequalities** — An inequality such as `|x−4| < 3` can be viewed by comparing the graph `y = |x−4|` with the horizontal line `y = 3`. The solution consists of the x-values where the absolute-value graph is below the horizontal line. This visually explains why the answer is the interval between the two intersection points. 23. **Why `<` absolute-value problems give one interval while `>` problems give two** — “Less than a distance” keeps x **inside** a boundary, producing one middle interval. “Greater than a distance” puts x **outside** the boundary, producing a left interval and a right interval. This is the geometric reason for the difference between **and** and **or** absolute-value inequalities. 24. **Modeling real-world comparisons with inequalities** — Inequalities can determine when one option becomes cheaper, larger, faster, or otherwise preferable to another. The procedure is to define a variable, write expressions for the competing quantities, place the proper inequality between them, solve it, and interpret the result in the context of the problem. 25. **Checking a real-world answer in context** — After solving an applied inequality, test a value from the proposed solution in the original situation. The toll example on page 203 compares `20 + 3.25x` with `7.25x`, producing `x > 5`; testing six crossings confirms that the first pricing plan costs less. This verifies not merely the algebra but the interpretation of the answer.
