Linear Inequalities Pattern - Verify

This pattern tests whether a given point or set of values satisfies a linear inequality. The core skill is substitution: plug in the coordinates and check whether the resulting statement is true.

 

Substituting a Point into a Single Inequality

Given an inequality like $y \geq 3x - 5$, test a point $(x, y)$ by replacing $x$ and $y$ with the given numbers. If the resulting statement is true, the point is a solution.

Which point $(x, y)$ is a solution to the inequality $y \geq 3x - 5$?

A) $(2, 0)$
B) $(4, 1)$
C) $(2, 2)$
D) $(-1, -10)$

Test each option by substituting into $y \geq 3x - 5$.

Option C: $x = 2$, $y = 2$. The right side is $3(2) - 5 = 1$. Check: $2 \geq 1$. True. The answer is C.

The other options fail: A gives $0 \geq 1$ (false), B gives $1 \geq 7$ (false), D gives $-10 \geq -8$ (false).

The most common mistake is reversing the substitution — putting $y$ where $x$ goes or vice versa. Always match $(x, y)$ with the correct variable.

Which point $(x, y)$ is a solution to the inequality $4y + 8 > 2x$?

A) $(4, 0)$
B) $(2, 0)$
C) $(10, 0)$
D) $(0, -3)$

Test option B: $x = 2$, $y = 0$. Left side: $4(0) + 8 = 8$. Right side: $2(2) = 4$. Check: $8 > 4$. True. The answer is B.

Option A: $4(0) + 8 = 8$ vs. $2(4) = 8$. Check: $8 > 8$. False (strict inequality, not $\geq$). Watch out — when the inequality is strict ($>$ or $<$), the boundary value itself does not satisfy it.

When the inequality has a fraction, compute both sides carefully:

Which ordered pair $(x, y)$ is a solution to $y < -\dfrac{1}{2}x + 1$?

A) $(4, -1)$
B) $(0, 2)$
C) $(-2, 3)$
D) $(4, -3)$

Test option D: $x = 4$, $y = -3$. Right side: $-\dfrac{1}{2}(4) + 1 = -2 + 1 = -1$. Check: $-3 < -1$. True. The answer is D.

Option A: right side is also $-1$. Check: $-1 < -1$. False — the value equals the boundary, but the inequality is strict.

 

Quadrant Identification

A simple variant gives you a point and asks which system of sign-based inequalities it satisfies. The four quadrants correspond to:

Quadrant I: $x > 0$ and $y > 0$ (both positive) Quadrant II: $x < 0$ and $y > 0$ Quadrant III: $x < 0$ and $y < 0$ (both negative) Quadrant IV: $x > 0$ and $y < 0$

Which system of inequalities has the point $(-1, -9)$ as a solution?

A) $x > 0$ and $y > 0$
B) $x > 0$ and $y < 0$
C) $x < 0$ and $y > 0$
D) $x < 0$ and $y < 0$

Since $-1 < 0$ and $-9 < 0$, both coordinates are negative. This matches Quadrant III: $x < 0$ and $y < 0$. The answer is D.

Just check the sign of each coordinate. No computation needed.

 

Verifying a Point in a System of Two Inequalities

When a system has two inequalities, the point must satisfy both. Substitute into each inequality separately and confirm both statements are true.

$y + 3x \leq 9$ $y - x \geq 1$

Which point $(x, y)$ is a solution to the given system?

A) $(1, 4)$
B) $(3, 2)$
C) $(-1, -2)$
D) $(0, 10)$

Test option A: $x = 1$, $y = 4$.

First inequality: $4 + 3(1) = 7 \leq 9$. True. Second inequality: $4 - 1 = 3 \geq 1$. True.

Both satisfied. The answer is A.

Option D fails the first: $10 + 3(0) = 10 \leq 9$ is false. Option B fails the second: $2 - 3 = -1 \geq 1$ is false.

$y \leq 4x + 10$ $y \geq -2x - 8$

Which point $(x, y)$ is a solution to the given system?

A) $(-2, 3)$
B) $(-1, 5)$
C) $(-5, 0)$
D) $(1, -12)$

Test option B: $x = -1$, $y = 5$.

First: $5 \leq 4(-1) + 10 = 6$. True. Second: $5 \geq -2(-1) - 8 = -6$. True.

Both satisfied. The answer is B.

Option A: First gives $3 \leq 4(-2) + 10 = 2$, which is $3 \leq 2$ — false.

Sometimes you're given one coordinate and must find the other. Substitute the known value and solve:

$y \leq 0$ $8y - 5x > 25$

The point $(x, -10)$ is a solution. Which could be $x$?

A) $-21$
B) $-22$
C) $-20$
D) $0$

First inequality: $-10 \leq 0$. True for any $x$.

Second inequality: $8(-10) - 5x > 25$, so $-80 - 5x > 25$, meaning $-5x > 105$, thus $x < -21$.

Of the options, only $x = -22$ satisfies $x < -21$. The answer is B.

 

Table Verification

Some questions give an inequality and four tables of $(x, y)$ pairs. You must find the table where every row satisfies the inequality. The strategy: compute the boundary value for each $x$, then compare.

$y > 25x + 150$

For which table are all values of $x$ and $y$ solutions to the given inequality?

$x$	$y$ (A)	$y$ (B)	$y$ (C)	$y$ (D)
10	400	390	405	410
20	650	640	650	660
30	900	890	905	910

Compute the boundary $25x + 150$ for each $x$:

$x = 10$: boundary $= 400$. Need $y > 400$. Options A and B fail (400 and 390). $x = 20$: boundary $= 650$. Need $y > 650$. Option C fails at this row (650 is not $> 650$). $x = 30$: boundary $= 900$.

Only table D has all values strictly above the boundary: $410 > 400$, $660 > 650$, $910 > 900$. The answer is D.

$P \leq \dfrac{1}{2}t + 5$

The profit $P$, in thousands of dollars, of a startup is modeled by the inequality above, where $t$ is the number of months since launch. For which table are all values of $t$ and $P$ solutions?

$t$	$P$ (A)	$P$ (B)	$P$ (C)	$P$ (D)
10	9	10	11	10
20	16	14	16	15
30	18	19	21	22

Compute boundary $\dfrac{1}{2}t + 5$:

$t = 10$: $10$. Need $P \leq 10$. Options A (9) and B (10) and D (10) pass; C (11) fails. $t = 20$: $15$. Need $P \leq 15$. A (16) and C (16) fail; B (14) and D (15) pass. $t = 30$: $20$. Need $P \leq 20$. B (19) passes; D (22) fails.

Only table B satisfies all three rows. The answer is B.

For table problems, it's efficient to start with the row that has the tightest constraint. One failure eliminates the entire table.

 

System Table Verification

A harder variant gives a system of inequalities and asks which table satisfies both. Each row must pass every inequality.

$y > 2x$ $x > 5$

For which table are all values solutions to the system?

$x$	$y$ (A)	$y$ (B)	$y$ (C)	$y$ (D)
6	12	13	9	5
7/8	10	17	11	8
9/10	15	21	5	1

First check $x > 5$ for all rows. Tables C and D include $x = 4$ or lower — but actually all tables use $x$ values of 6, 7 or 8, and 9 or 10. Tables C uses $x = 4$ and D uses $x = 3$ — eliminated.

For table A: $y > 2x$ gives $12 > 12$ (false at $x = 6$). Eliminated.

For table B: $13 > 12$, $17 > 16$, $21 > 20$. All true. The answer is B.

 

Key Strategies

You don't need to test every option. Start with the option that looks most likely to succeed, or eliminate options quickly by finding a single row that fails. For strict inequalities ($>$ or $<$), remember that boundary values do not satisfy the inequality. For non-strict ($\geq$ or $\leq$), boundary values do count.