Digital SAT · Math · Unit 1 · Algebra

Lesson 1.5 · Linear Inequalities

On the test: 1–3 questions a test — word problems and graphs more than pure solving

Big idea. An inequality is a balance that is allowed to tip one way. You solve it with exactly the same moves as an equation, with ONE extra rule: multiplying or dividing by a negative number flips the sign. Its answer is not one number but a whole range — on a number line a ray, on the plane a shaded half.

🎛 Move it · why the sign flips

$2 < 5$ . Multiply both by $k$ . Slide $k$ below zero and watch the order of the two numbers swap.

-15-10-505101525before: 2 < 5

multiply both by k = -2

-15-10-5051015-4-10after × -2: -4 > -10

Negative multiplier: the line is mirrored, the smaller number becomes the bigger — so the sign must FLIP.

🧑‍🏫 Rule 1 · Solve like an equationstep 1 of 3

1
the question
$4 h \geq 24$
Practice Test 9: walk at least 24 km at 4 km/h. Least hours? Adding, subtracting, and multiplying/dividing by POSITIVE numbers work exactly as before.

🧑‍🏫 Rule 2 · Multiply or divide by a negative → FLIPstep 1 of 5

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📋 The question · SAT · October US 2024 · Math 800 Book Q31
If $x \geq - 6$ represents all solutions to $a x - 27 \leq 15$ , where $a$ is a constant, what is the greatest possible value of $a$ ?
(type the answer)
the question
$a x - 27 \leq 15 has solutions x \geq - 6$
October US 2024: find the greatest possible $a$ .

🧑‍🏫 Rule 3 · Between two values: a compound inequalitystep 1 of 4

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📋 The question · SAT · June 2023 · Math 800 Book Q3
In December 2017, the lowest temperature recorded in a certain city was 40°F and the highest was 90°F. Which inequality is true for all values of $t$ , where $t$ represents any temperature, in °F, recorded in the city in December?
1. A) $40 \leq t \leq 90$
2. B) $t \leq 40$
3. C) $t \leq 50$
4. D) $t \geq 90$
the question
$lowest 40, highest 90$
June 2023: every December temperature $t$ was between the lowest and the highest.

Why it works · why the flip

- 2 > - 5

even though

2 < 5

: on the negative side, the number closer to zero is the BIGGER one. Multiplying by a negative mirrors the number line, so the order — and the sign — reverses.

🧑‍🏫 Rule 4 · The vocabularystep 1 of 5

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📋 The question · SAT · December US 2024 · Math 800 Book Q37
A team earns $12 per coupon book and wants at least $1,320. Which inequality describes $n$ ?
1. A) $n + 12 \leq 1320$
2. B) $n + 12 \geq 1320$
3. C) $12 n \leq 1320$
4. D) $12 n \geq 1320$
at least
$\geq$
“At least 1,320” = 1,320 or more (December US 2024: $12 n \geq 1320$ ).

🧑‍🏫 Rule 5 · Whole-number answers: round the RIGHT waystep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q13
A truck can tow a trailer if the combined weight of the trailer and its packages is no more than 4,600 pounds. What is the maximum number of packages it can tow in a 600-pound trailer if each package weighs 120 pounds?
1. A)33
2. B)34
3. C)38
4. D)39
the question
$600 + 120 n \leq 4600$
March 2024: a truck tows at most 4,600 lb; trailer 600 lb, packages 120 lb each. Maximum packages?

🧑‍🏫 Rule 6 · Words that hide two inequalitiesstep 1 of 4

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📋 The question · SAT · November 2024 · Math 800 Book Q33
Antwan charges $10.00 plus $8.50 per hour; Taylor charges $7.00 plus $9.50 per hour. For which values of $x$ is Taylor’s total charge greater than Antwan’s?
1. A) $2 \leq x \leq 3$
2. B) $3 \leq x \leq 4$
3. C) $x < 2$
4. D) $x > 3$
the question
$7 + 9.5 x > 10 + 8.5 x$
November 2024: when is Taylor’s charge greater than Antwan’s?

Trap · the wrong rounding

For “maximum” round DOWN even at .9; for “minimum / least” round UP even at .1. Always check your number in the original condition.

🎛 Move it · shade it, test it

Choose the sign, slope and intercept. Move the test point: the plane shades the solutions, and the box tells you whether the point makes the inequality TRUE.

y ≤ -1x+3

y \leq - 1 x + 3

slope m-1b3

test point

x1y1

1 ≤ -1·1 + 3 = 2? TRUE → the point is a solution (in the shaded part).

“<” or “≤” shades BELOW the line; “>” or “≥” shades ABOVE. With ≤ / ≥ the line itself counts (solid); with < / > it does not (dashed on the test).

🧑‍🏫 Rule 7 · Read a shaded graph in three looksstep 1 of 4

1
look 1: the line
$y = m x + b$
The boundary is an ordinary line — find its slope and intercept from two grid points.

🧑‍🏫 Rule 8 · Is this point (or table) a solution? Plug it instep 1 of 4

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📋 The question · SAT · August US 2023 · Math 800 Book Q4
For $y \leq \frac{2}{3} x + 5$ , which table has all values of $x$ and $y$ as solutions? A: $(- 1.5, 6), (1.5, 5), (3, - 4)$ B: $(- 1.5, 5), (1.5, 7), (3, 8)$ C: $(- 1.5, 6), (1.5, 5), (3, 6)$ D: $(- 1.5, 3), (1.5, 5), (3, 6)$
1. A)A
2. B)B
3. C)C
4. D)D
the question
$y \leq \frac{2}{3} x + 5$
August US 2023: which table has ALL its points as solutions?

🧑‍🏫 Rule 9 · A constant from the graphstep 1 of 4

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📋 The question · SAT · March 2024 · Math 800 Book Q11
$figure from the test$
The shaded region represents the solutions to $- 18 y < c$ , where $c$ is a constant. What is $c$ ?
1. A)162
2. B)9
3. C) $- 9$
4. D) $- 162$
the question
$- 18 y < c$
March 2024: the graph shades everything above the dashed line $y = - 9$ .

Real question 1SAT · March 2024 · Math 800 Book Q11

$figure from the test$

The shaded region shown represents the solutions to the inequality $- 18 y < c$ , where $c$ is a constant. What is the value of $c$ ?

🧑‍🏫 Rule 10 · A solution passes all of themstep 1 of 3

1
📋 The question · SAT · March 2024 · Math 800 Book Q12
$y \leq x + 2$ and $y \geq - 3 x - 4$ . Which point is a solution?
1. A) $(0, - 8)$
2. B) $(0, 8)$
3. C) $(- 8, 0)$
4. D) $(8, 0)$
the question
$y \leq x + 2, y \geq - 3 x - 4$
March 2024: which point is a solution of the system?

🧑‍🏫 Rule 11 · Squeeze the x-valuesstep 1 of 4

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📋 The question · SAT · December 2024 · Math 800 Book Q40
$y < 42 - 7 x$ and $\frac{y}{7} > 10$ . Which inequality represents the $x$ -values of all solutions?
1. A) $x > 6$
2. B) $x < 6$
3. C) $x > - 4$
4. D) $x < - 4$
the question
$y < 42 - 7 x, \frac{y}{7} > 10$
December 2024: which inequality gives the $x$ -values of all solutions?

🧑‍🏫 Rule 12 · Which region holds no solutions?step 1 of 3

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📋 The question · SAT · June 2023 · Math 800 Book Q2
In the $x y$ -plane, which of the following does NOT contain any points $(x, y)$ that are solutions to $7 x + 4 y > 12$ ?
1. A)The region where $x > 0$ and $y > 0$
2. B)The region where $x < 0$ and $y > 0$
3. C)The region where $x < 0$ and $y < 0$
4. D)The region where $x > 0$ and $y < 0$
the question
$7 x + 4 y > 12$
June 2023: which region contains NO solutions?

Desmos move · see the overlap

Type each inequality on its own line. Desmos shades each one; the solutions of the system are where ALL the shadings overlap. Tap a point there to read it.

🧑‍🏫 Rule · |x − a| < k: within k of astep 1 of 3

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📋 The question · Gen Z Study
Which inequality is equivalent to $∣ x - 4∣ < 3$ ?
1. A) $1 < x < 7$
2. B) $x < 1$ or $x > 7$
3. C) $- 3 < x < 3$
4. D) $x < 7$
meaning
$∣ x - 4∣ < 3$
“ $x$ is less than 3 away from 4.”

🧑‍🏫 Rule · |x − a| > k: farther than k from astep 1 of 3

1
📋 The question · Gen Z Study
Which values of $x$ satisfy $∣2 x + 1∣ \geq 7$ ?
1. A) $- 4 \leq x \leq 3$
2. B) $x \leq - 4$ or $x \geq 3$
3. C) $x \geq 3$
4. D) $x \leq - 3$ or $x \geq 4$
split
$2 x + 1 \geq 7 or 2 x + 1 \leq - 7$
Greater-than → two separate rays, joined by “or”.

Take with you

▸Same moves as equations; multiplying or dividing by a NEGATIVE flips the sign.
▸At least $\geq$ , at most / no more than $\leq$ , more/less than $>$ / $<$ .
▸Maximum → round DOWN; minimum → round UP; then check in the original condition.
▸A range times a count: multiply BOTH ends.
▸Graph: solid for $\leq / \geq$ , dashed for $< / >$ ; above means $y >$ , below means $y <$ . Test a point by plugging it in.
▸A system’s solution must satisfy EVERY inequality; chain them to squeeze out a range.

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