Solve it like an equation, flip for a negative, then decide which way to round.
A food truck pays a one-time permit fee of 180 dollars and spends 6.25 dollars on ingredients for each meal it sells. The owner needs the total cost to stay under 500 dollars. What is the greatest number of meals the truck can make?
180 + 6.25m < 500
6.25m < 320, so m < 51.2. A meal count must be a whole number, and the greatest whole number below 51.2 is 51: 180 + 6.25(51) = 498.75, still under 500. C rounds 51.2 up, and 52 meals cost 505 dollars — over budget. D divides 500 by 6.25 and never subtracts the permit fee. A backs off one extra meal because the sign is strict, but 51.2 is not a whole number, so nothing is lost at the edge.
1
Turn the phrase into a symbol before you touch the algebra.
"At least" and "at most" include the boundary. "More than" and "fewer than" do not.
The stem says
Write
Is 12 itself allowed?
at least 12
x ≥ 12
yes
no less than 12
x ≥ 12
yes
at most 12
x ≤ 12
yes
no more than 12
x ≤ 12
yes
a maximum of 12
x ≤ 12
yes
more than 12
x > 12
no
fewer than 12
x < 12
no
under 12
x < 12
no
a one-time permit fee of 180 dollars
180, a plain number
6.25 dollars for each of m meals
6.25m, a coefficient
stay under 500 dollars
< 500, and 500 itself is out
180 + 6.25m < 500
The SAT writes the cost sentence exactly the way it writes it for a linear
equation. Only the last three words change — and they choose the symbol.
2
Solve it exactly as you would an equation.
Adding and subtracting never disturb the direction of the sign.
− 180 both sides÷ 6.25 both sides
180 + 6.25m < 500
the inequality from step 1
6.25m < 320
500 − 180 = 320
m < 51.2
320 ÷ 6.25 = 51.2, and 6.25 is positive
m < 51.2
Desmos does the ugly division
The Digital SAT keeps a graphing calculator on screen the whole time. Type
180 + 6.25x < 500 into Desmos and it shades the values of x
that work, up to a boundary it will label for you; or just divide 320 by 6.25 on
the keypad. Decimals like 51.2 and 6.25 are normal on this test — they are not a
sign that you went wrong.
3
Multiplying or dividing by a negative turns the sign around.
This is the only rule that separates an inequality from an equation. Adding a negative does nothing; multiplying by one reverses everything.
2 started on the left of 5. After multiplying by −1, −2 is on the right of −5.
The two numbers swapped places, so the sign between them has to swap as well.
−4x + 9 ≥ 33
an SAT-style stem
−4x ≥ 24
− 9 both sides, sign untouched
x ≤ −6
÷ (−4) both sides — the sign flips
The other route: never divide by a negative at all.
−4x + 9 ≥ 33
the same stem
9 ≥ 33 + 4x
+ 4x both sides, so x lands positive
−24 ≥ 4x, so −6 ≥ x
÷ 4, no flip needed
The forgotten flip is the distractor
Skip it here and you get x ≥ −6, which is waiting in the choices. One test number
settles it: x = 0 gives 9 ≥ 33, false — so 0 cannot be in the answer, and x ≥ −6
would let it in. Only × and ÷ by a negative flip the sign; subtracting 9 never does.
4
A compound inequality is one problem with three parts.
Whatever you do, do it to all three — and keep each end's sign exactly where it started.
−5 <
2x + 3 ≤
11
lower wall · the middle · upper wall
−5 < 2x + 3 ≤ 11
the given
−8 < 2x ≤ 8
− 3 from all three parts
−4 < x ≤ 4
÷ 2 in all three parts
The strict end does not wander
The < started on the left and stays on the left, so the answer is −4 < x ≤ 4,
never −4 ≤ x < 4. The one thing that does move both signs is dividing
all three parts by a negative: from −9 < −3x ≤ 6 you get 3 > x ≥ −2, which is
read back as −2 ≤ x < 3.
5
"Greatest" rounds down. "Least" rounds up. Decide it on purpose.
The algebra hands you a decimal. Which whole number you hand back is a separate decision — and it is the one the SAT is actually testing.
The algebra gives
The question wants
The answer
m < 51.2
the greatest integer
51 — step back to the last whole number
x > 19.4
the least integer
20 — step forward to the next one
x > 8
the least integer
9 — 8 is excluded by the strict sign
x ≥ 8
the least integer
8 — ≥ lets the boundary in
Rounding 51.2 the way you round a measurement happens to give 51 as well. Rounding
19.4 that way gives 19 — and 19 is not in the solution set at all. The decimal does
not decide; the word "greatest" or "least" does.
The sign only matters when the boundary is already a whole number
7x + 4 > 60 gives x > 8, landing exactly on an integer. Because the sign is
strict, 8 is out and the least integer is 9. Had the stem read 7x + 4 ≥ 60, the
answer would be 8. Check the sign in that case only — the rest of the time the
decimal has already excluded the boundary for you.
6
Two costs compared is still one inequality.
"For which values is A cheaper than B" means putting the two expressions on opposite sides of the right symbol.
A: 64 + 9h · B: 145 + 4h
two equipment quotes for h hours
64 + 9h < 145 + 4h
"A costs less than B"
5h < 81
− 64 and − 4h both sides
h < 16.2
÷ 5, positive, so no flip
A is cheaper for any job shorter than 16.2 hours
At h = 16.2 the two quotes are equal, at 209.80 dollars each: 64 + 145.80 and
145 + 64.80. That single equality is the boundary, and the word "cheaper" decides
which side of it you keep. Desmos will graph y = 64 + 9x and y = 145 + 4x and
label the crossing if you would rather see it than solve it.
✓
Check: test a number on each side of the boundary.
One substitution catches a missed flip, and one more catches a bad round.
m = 51180 + 6.25(51) = 498.75under 500 ✓ the answer
m = 52180 + 6.25(52) = 505over budget ✗
x = −6−4(−6) + 9 = 33, and 33 ≥ 33the edge is included by ≥ ✓
x = 09 ≥ 33 is falseso x ≤ −6, not x ≥ −6 ✓
x = 42(4) + 3 = 11, and 11 ≤ 11the closed end holds ✓
Remember the flow
Read the phrase · Solve · Flip for a negative · Round on purpose · Test
The arithmetic gives a decimal; the question decides which whole number to hand in.
Now you try
Practice problem
A shipping company charges 8.50 dollars per package plus 1.75 dollars per pound. The charge for one package was at least 42 dollars. What is the least possible whole number of pounds the package could weigh?
8.50 + 1.75w ≥ 42
1.75w ≥ 33.5, so w ≥ 19.142…, and the question asks for the least whole number that works — round up to 20. Check: 8.50 + 1.75(20) = 43.50, which is at least 42 ✓, while 19 pounds gives only 41.75 ✗. A rounds 19.14 down, the way you would round a measurement, but 19 is not in the solution set at all. C divides 42 by 1.75 and forgets the 8.50. D adds the 8.50 instead of subtracting it: 50.5 ÷ 1.75 = 28.86, rounded up to 29.
Hint: "at least 42" is ≥, so the total is allowed to equal 42.
Subtract the fixed 8.50 first, divide by 1.75 second, and then read the words
"least possible" one more time before you pick.
