So far every equation has had one exact answer. But most real decisions are not about exact amounts — they're about enough, too much, or at least. "Copper needs at least 8 kg of hay per day." "The trailer can carry at most 2 horses." "Mia needs more than 50 € to enter." These are inequalities — they describe a range of answers, not a single point.
The Four Symbols
Symbol
Meaning
Example
In words
$>$
greater than
$x > 3$
x is more than 3
$<$
less than
$x < 10$
x is less than 10
$\geq$
greater than or equal to
$x \geq 5$
x is at least 5
$\leq$
less than or equal to
$x \leq 8$
x is at most 8
Open circle vs. filled circle on a number line:
● (filled) = the value is included (use with $\geq$ and $\leq$)
○ (open) = the value is not included (use with $>$ and $<$)
Worked Examples
Worked Example 1 — reading and graphing
Mia needs at least 40 € to enter the competition. Let $m$ = her savings. Write and graph the inequality.
Inequality: $m \geq 40$
On a number line: draw a filled circle at 40, shade everything to the right (all values 40 or more).
Filled circle because 40 itself is allowed (she can enter with exactly 40 €).
Worked Example 2 — solving a one-step inequality
Mia's weekly savings must be enough so that in 6 weeks she will have over 90 €. How much must she save each week?
Let $s$ = weekly savings:
$$6s > 90 \quad \mid \div 6$$
$$s > 15$$
She must save more than 15 € per week. (15 € exactly is not enough — she needs strictly more than 90 € at the end.)
Solving an inequality works exactly like solving an equation — same moves, same | notation — as long as you don't divide by a negative. (More on that in Lesson 07.)
Worked Example 3 — two-sided situation
Show rules say a horse may not weigh less than 300 kg to compete in the senior class. Copper weighs 480 kg. Does he qualify? What is the constraint as an inequality?
Constraint: $w \geq 300$ kg. Copper: $480 \geq 300$ ✓ — he qualifies.
Practice Problems
Problems 1–3: write the inequality from the description and graph it (draw on the number line box).
Hay order. The stable requires each owner to keep at least 20 kg of hay in stock at all times.
Let $h$ = kilograms of hay Mia has. Write the inequality:
Graph on number line (mark the key value, show direction, open or filled circle):
Weight limit. The horse trailer can carry at most 1200 kg total.
Let $w$ = total weight loaded. Write the inequality:
Under par. To qualify for the finals, a rider's score must be strictly above 70.
Let $s$ = score. Write the inequality:
Problems 4–7: solve the inequality and graph the solution.
Training sessions. $x + 3 > 10$
Solve:
Graph:
Vet recommendation. $x - 5 \leq 12$
Solve:
Graph:
Feed bags. Copper eats 4 kg of grain per day. Mia wants the bag to last more than 10 days. The bag weighs $x$ kg. Write and solve the inequality.
Graph:
Savings goal. Mia earns 15 € per week. She needs at least 90 € to buy a new riding helmet. Write and solve the inequality for the number of weeks $w$ she must save.
Graph:
From here — write the equation yourself, then solve.
Competition weight class. The junior class requires a horse to weigh strictly less than 500 kg. Copper weighs 480 kg now and is gaining 2 kg per week. How many more weeks until he no longer qualifies for juniors?
Write the inequality, solve for $w$, and interpret the answer.
Graph:
Budget check. Mia has $80 in her account. She needs to pay a farrier (a specialist who trims hooves). The farrier charges $x per visit plus a $12 call-out fee. Mia can afford at most $80 total. What is the maximum she can pay per visit?
What's next → You've solved one-step inequalities using the same moves as equations. Lesson 07 takes this one step further with two-step inequalities — and reveals the one important difference from equations: when you divide or multiply both sides by a negative number, the inequality direction flips. $-2x > 8$ becomes $x < -4$, not $x > -4$. We'll see exactly why, and when it matters.
Show answers
$h \geq 20$; filled circle at 20, shaded right
$w \leq 1200$; filled circle at 1200, shaded left
$s > 70$; open circle at 70, shaded right
$x + 3 > 10$ → $x > 7$; open circle at 7, shaded right
$x - 5 \leq 12$ → $x \leq 17$; filled circle at 17, shaded left
$4x > 40$ → $x > 10$ kg; the bag must weigh more than 10 kg
$15w \geq 90$ → $w \geq 6$ weeks
$480 + 2w < 500$ → $2w < 20$ → $w < 10$ weeks; so after 9 more weeks he still qualifies, but from week 10 he does not