8.3 Inequalities & Translating Word Problems

Solving Linear Inequalities

A linear inequality in one variable uses one of four symbols—< (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to)—instead of an equals sign. Solving an inequality is almost identical to solving an equation: you apply the same inverse operations to both sides. There is one critical exception.

The golden rule of inequalities: When you multiply or divide both sides by a negative number, reverse the direction of the inequality sign.

Why does the sign flip? Consider a true statement: −2 < 3. Multiply both sides by −1 without flipping, and you get 2 < −3, which is false. With the flip, you get 2 > −3, which is true. Multiplying or dividing by a negative reverses the order of the numbers on the number line, so the inequality must reverse to stay true.

Adding or subtracting a negative does NOT flip the sign—only multiplication or division by a negative does.

Example 1. Solve x + 5 < 12.

Subtract 5 from both sides: x < 7. The solution is any value of x less than 7. No flip was needed because we did not multiply or divide by a negative.

Example 2. Solve −3x > 9.

Divide both sides by −3. Because you are dividing by a negative, reverse the sign: x < −3. The solution is any x less than −3.

Example 3. Solve 2x − 4 ≤ 10.

Add 4 to both sides: 2x ≤ 14. Divide by 2 (positive, no flip): x ≤ 7.

Example 4. Solve −2x + 6 ≥ 14.

Subtract 6 from both sides: −2x ≥ 8. Divide by −2 and flip the sign: x ≤ −4. Notice the flip turned ≥ into ≤.

Graphing Inequalities on a Number Line

The solution to an inequality is a range of values, not a single number. You can show this range on a number line:

• An open circle ○ marks a value that is NOT included. Use it for < and >.
• A closed circle ● marks a value that IS included. Use it for ≤ and ≥.

Then shade the part of the number line that contains the solution values. For x < 7, draw an open circle at 7 and shade everything to the left. For x ≤ −4, draw a closed circle at −4 and shade everything to the left (because the solution is values less than or equal to −4).

Example 5. Graph x ≥ 3.

Closed circle at 3, shaded to the right (all values greater than or equal to 3).

Example 6. Graph x < −2.

Open circle at −2, shaded to the left (all values strictly less than −2).

Translating English into Algebra

Many NEX word problems describe a relationship in English and ask you to write an expression or equation. Learning the standard translations speeds this up.

Important order note: "7 less than a number" translates to x − 7, not 7 − x. The phrase "less than" reverses the order of the words you see. The number being subtracted from is the one mentioned first ("a number"), and the amount being subtracted is the one mentioned second ("7"). This is one of the most common translation errors on the NEX.

Similarly, "10 more than a number" is x + 10, not 10 + x—though here order does not matter for addition, the variable should be the subject of the sentence.

Setting Up and Solving a Word Problem

Use a consistent four-step process:

1. Identify the unknown and choose a variable to represent it.
2. Write an equation from the relationships in the problem.
3. Solve the equation.
4. Check that the answer makes sense in the original context.

Example 7 (per-rate problem). A nursing assistant earns $18 per hour. If she earned $414 this week, how many hours did she work?

Let h = hours worked. The pay is rate times hours: 18h = 414. Divide by 18: h = 23. Check: 18·23 = 414. She worked 23 hours.

Example 8 (consecutive integers). The sum of two consecutive integers is 37. Find them.

Let the smaller integer be n; the next is n + 1. Their sum is n + (n + 1) = 37, so 2n + 1 = 37. Subtract 1: 2n = 36. Divide by 2: n = 18. The integers are 18 and 19. Check: 18 + 19 = 37.

Example 9 (inequality word problem). A clinic allows a nurse to administer at most 30 doses per shift. She has already given 12. How many more can she give?

Let x = additional doses. "At most" means ≤. The total is 12 + x, so 12 + x ≤ 30. Subtract 12: x ≤ 18. She can give at most 18 more doses.

Example 10 (full translation). A patient's heart rate is 8 beats per minute above the normal resting rate of 72. Write and solve an equation.

Let h = heart rate. "Above" means addition: h = 72 + 8 = 80. The heart rate is 80 beats per minute.

Common Traps

• Not flipping the inequality sign on a negative multiply or divide. If you solve −2x > 10 and write x > −5, you are wrong; the correct answer is x < −5. Always check whether your last step multiplied or divided by a negative.
• Misreading "less than" order. "7 less than x" is x − 7, but "7 is less than x" is 7 < x. The two phrases look similar but mean different things. Read carefully: if a number appears before "less than," it is usually the amount being subtracted.
• Using the wrong inequality symbol. "At most" means ≤ (closed circle). "Fewer than" means < (open circle). Match the words to the symbol before solving.
• Forgetting that a solution is a range. An inequality answer is a set of values, not a single number. If the question asks for "all possible values," give the inequality, not one example.