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Stage 8 · Advanced Algebra Skills
Writing Equations & Inequalities from Word Problems
On the GED, many word problems aren't asking you to solve — they're asking "which equation matches this situation?" This lesson is about reading the story, finding the rate and the starting value, and setting up the right equation or inequality.
This stage contains advanced algebra skills that build on earlier lessons. You should already feel comfortable with equations, inequalities, slope, and y = mx + b before starting this page. This lesson combines algebra, reading comprehension, and problem solving — and you should already feel comfortable with the simpler translation problems from earlier lessons.
You already know most of these ideas. A word problem is just a story that describes an algebra relationship. The rate is usually the slope. The starting amount is usually b. Your job is to read carefully, find those two numbers, decide the sign, and pick the equation or inequality that matches.
The GED often tests equation setup more than solving. A huge number of word problems just ask "which equation matches this situation?" If you can set it up, you've earned the point — no solving required. Focus on what is changing, and find the starting amount.
It's Really Just y = mx + b
Many GED word problems follow the exact same structure as y = mx + b — the line equation you already know. Once you see that, every problem looks familiar.
y = mx + b
m = the rate (how fast it changes) · b = the starting value (where it begins) · x = what changes over time
The rate is usually the slope. The starting amount is usually b. Watch for words like per, each, every, hourly, daily, weekly, and monthly — they almost always describe the rate. A one-time amount (a fee, a deposit, a starting height) is usually the starting value.
5 Questions to Ask Every Time
Before writing anything, read the story and answer these five questions. They turn any word problem into a setup.
1
What is changing? That's your variable (often x). It's the thing that grows or shrinks — hours, miles, months, tickets.
2
Is it increasing or decreasing? Positive means increasing. Negative means decreasing.
3
What is the starting value? The amount before anything changes — a fee, a deposit, a starting height. That's b.
4
What do the variables represent? Say it in words: "x = number of hours," "C = total cost." Units keep you honest.
5
Is this an equation or an inequality? Words like at least, at most, no more than, or budget signal an inequality.
Signs matter — read the direction of the change. Gaining money → positive. Losing altitude → negative. Draining water → negative. Underground values → negative. Growing savings → positive. Decide the sign before you write the rate.
Inequality Vocabulary
Inequalities represent limits and restrictions — a budget, a maximum, a minimum, a goal. The words tell you which symbol to use. Learn these phrases and the matching sign.
The words say…
Symbol
Means
at least, minimum, no less than
≥
greater than or equal to
at most, maximum, no more than
≤
less than or equal to
more than, must exceed, greater than
>
strictly greater than
less than, under, stay below, fewer than
<
strictly less than
"or equal to" is the tell.At least and at most include the boundary, so they use ≥ or ≤. More than and less than do not include it, so they use > or <.
Getting the Inequality Direction Right
Here's the most common trap: students understand the situation perfectly, then write the inequality backwards. The fix is to remember what an inequality really does.
An equation
Says two things are equal. It describes one value: "A person is 14 years old."
An inequality
Compares one quantity to another: "Jordan is older than Mia." A comparison, not a single value.
🎂 The Age Analogy
"Jordan is older than Mia" can be written two correct ways — and both mean exactly the same thing:
Jordan > Mia
Jordan's age is greater
Mia < Jordan
Mia's age is less
Notice what happened: the quantities switched places, and the inequality direction switched too. The relationship stayed identical. Think in words first, then write the symbols.
Because both directions are correct, rewrite the comparison in whichever direction sounds most natural to you. One version is usually easier to picture than the other.
Harder to picture
money > spending · goal > actual
Easier to picture
spending < money · actual < goal
Match the Words, Then the Numbers
Once you've decided the comparison with words, plug the numbers into the quantities. Compare the words first — then match the numbers to the quantities.
1
Decide which quantity is larger or smaller — in plain words.
2
Pick the direction that feels easiest to understand.
3
Match the actual numbers to those quantities.
4
Choose the matching answer choice — the symbols should match the relationship between the words.
Words first — a budget example
A student has $50 total and spends $8 per movie ticket (m = tickets). Spending should stay below the total money available.
→ In words: spending should stay below the money available. So: spending < 50.
→ Now replace "spending" with the expression: spending = 8m.
8m < 50
🎯 GED Multiple Choice Strategy
Many GED word problems can be narrowed down before you fully set anything up. The trick: look at the structure of the answer choices first. Most look like y ? mx + b — a rate, a starting value, and a comparison.
The 3-step narrowing routine
1Identify the structure. Find the changing rate and the starting amount, then match those to the answer choices. This alone often removes two wrong answers.
2Decide the direction. Use words first — which quantity should be larger? Which amount has to stay below the other?
3Match words to symbols. The GED loves to reverse the comparison order. "spending < budget" and "budget > spending" mean the same thing — don't get tricked.
Identifying m and b is often the fastest GED strategy. First find the rate. Then find the starting amount. That alone often removes two wrong answers before you do any real thinking about direction. The answer choices can help you — eliminate impossible signs first.
Walkthrough: gym membership inequality
A gym charges a $25 starting fee and $12 per class. A student can spend no more than $100 (c = classes).
Step 1 — structure: rate = 12, starting value = 25, so the cost is 12c + 25. That eliminates any choice that doesn't have both 12 and 25. Step 2 — direction: total spending should stay at or below 100. In words: spending ≤ 100. Step 3 — match: 12c + 25 ≤ 100. And recognize that 100 ≥ 12c + 25 means the exact same thing — the GED may print it that way.
Pick the symbol that matches the words.
Common GED Structures
Almost every GED word problem is one of three shapes. Learn to spot which one you're looking at.
Increasing (positive rate)
Wages, reading pages, saving money, miles driven, steps walked. The total grows → +rate.
Decreasing (negative rate)
Draining water, a descending airplane, loan balances, temperatures dropping, fuel remaining. The total shrinks → −rate.
Goal / target situations. Reaching a savings goal, paying off debt, hitting a step goal, filling or emptying a tank. These often become an inequality ("at least the goal") or a target-value equation ("equals the goal").
🔢 Worked Examples
Watch the same reading routine in every example: find the rate, find the starting value, decide the sign, then write the setup. Each example lines the situation up against y = mx + b — first the words, then the formula, then the actual numbers underneath.
Color key: rate (m) · starting value (b) · variable · negative / decreasing
total = rate × amount + starting amount
y = mx + b
This is the pattern every example below maps onto.
Example 1 — Positive Rate
A taxi charges a $5 starting fee plus $2.50 per mile. Which equation gives the total cost C for m miles?
→ What's changing? The miles → variable m.
→ Rate: 2.50 per mile (cost goes up → positive). Starting value: 5 (the one-time fee).
total=rate × amount+starty=mx+bC=2.50m+5
Final equation: C = 2.50m + 5
Example 2 — Negative Rate
A plane starts at 30,000 feet and descends 100 feet per minute. Which equation gives the altitude A after t minutes?
→ What's changing? The minutes → variable t.
→ Descending means altitude decreases, so the rate is −100. Starting value: 30,000.
total=rate × amount+starty=mx+bA=−100t+30,000
Final equation: A = −100t + 30,000 (the rate is negative because the plane is descending)
Example 3 — Goal Situation ("at least")
Juno has $30. Each ride costs $1.50. She wants to keep at least $10 left over. Write an inequality for the number of rides r.
→ Spending is the changing part: rate −1.50 per ride. Starting amount: 30.
→ Money left, in y = mx + b order (rate × rides, then starting amount):