Linear Inequalities Pattern - Interpret

This pattern asks you to connect an inequality to its meaning — either by interpreting what an ordered pair or term represents in context, or by reading a graph to identify the corresponding inequality.

 

Interpreting an Ordered Pair in Context

Given an inequality like $95a + 105b \leq 600$ and a scenario, you'll be asked what a specific ordered pair means. The key is to match each variable to its real-world quantity, substitute, and state what the inequality symbol means.

$95a + 105b \leq 600$

An athlete plans a snack of apples and bananas. Each apple ($a$) has 95 calories and each banana ($b$) has 105 calories. The inequality represents combinations with at most 600 calories. Which statement best interprets $(a, b) = (3, 2)$?

A) Eating 3 apples and 2 bananas gives a total calorie intake $\leq 600$.
B) Eating 2 apples and 3 bananas gives a total calorie intake $\leq 600$.
C) Eating 3 apples and 2 bananas gives a total calorie intake $\geq 600$.
D) Eating 2 apples and 3 bananas gives a total calorie intake $\geq 600$.

The ordered pair $(a, b) = (3, 2)$ means $a = 3$ and $b = 2$: 3 apples and 2 bananas (not reversed). The inequality uses $\leq$, meaning "less than or equal to." Verify: $95(3) + 105(2) = 285 + 210 = 495 \leq 600$. True. The answer is A.

$8x + 6y \geq 150$

An event planner uses large tables ($x$) seating 8 and small tables ($y$) seating 6. The total must be at least 150 seats. What does $(x, y) = (12, 10)$ mean?

A) 10 large and 12 small tables give capacity $\leq 150$.
B) 12 large and 10 small tables give capacity $\geq 150$.
C) 12 large and 10 small tables give capacity $\leq 150$.
D) 10 large and 12 small tables give capacity $\geq 150$.

$x = 12$ large tables, $y = 10$ small tables. The symbol $\geq$ means "greater than or equal to." Total: $8(12) + 6(10) = 96 + 60 = 156 \geq 150$. True. The answer is B.

Common trap: swapping which variable is which. Always check: the first coordinate matches the first variable in the inequality.

$3.5v + 0.8p \leq 64$

A 64-GB flash drive stores video files ($v$, 3.5 GB each) and photo albums ($p$, 0.8 GB each). What does $(v, p) = (15, 10)$ mean?

A) 10 videos and 15 albums use $\geq 64$ GB.
B) 15 videos and 10 albums use $\geq 64$ GB.
C) 10 videos and 15 albums use $\leq 64$ GB.
D) 15 videos and 10 albums use $\leq 64$ GB.

$v = 15$ videos, $p = 10$ albums. The $\leq$ symbol means the storage used is at most 64 GB. Check: $3.5(15) + 0.8(10) = 52.5 + 8 = 60.5 \leq 64$. True. The answer is D.

 

Interpreting a Term in an Inequality

A harder version asks what a single term (like $20x$) means in context. The coefficient is the per-unit rate and the variable is the count, so their product is the total for that category.

$20x + 50y \geq 1500$

A warehouse ships small boxes ($x$, 20 lbs each) and large boxes ($y$, 50 lbs each). The total must be at least 1500 lbs. What does $20x$ represent?

A) The total weight of all small boxes.
B) The weight of each small box.
C) The total weight of all large boxes.
D) The weight of each large box.

The coefficient 20 is the weight per small box. The variable $x$ is the number of small boxes. So $20x = 20 \cdot x$ is the total weight of all the small boxes. The answer is A.

Option B describes just the coefficient (20), not the full term. Option C confuses small with large.

$45c + 30p \leq 240$

A caterer has 240 minutes of oven time. Chicken batches ($c$) take 45 minutes each; pasta pans ($p$) take 30 minutes each. What does $45c$ mean?

A) Total cooking time for all pasta.
B) Total cooking time for all chicken.
C) Cooking time per pan of pasta.
D) Cooking time per batch of chicken.

$c$ = number of chicken batches, 45 = minutes per batch. So $45c$ = total cooking time for all chicken. The answer is B. Option D describes just the coefficient.

 

Reading a Graph: Identifying the Inequality

When a graph shows a shaded region with a boundary line, you need to determine the inequality. Follow these steps:

Step 1. Find the boundary line equation $y = mx + b$. Read the $y$-intercept ($b$) and use two points to compute the slope ($m$).

Step 2. Determine strict vs. non-strict: a dashed line means $<$ or $>$ (strict); a solid line means $\leq$ or $\geq$.

Step 3. Determine the direction: if the region above the line is shaded, use $>$ or $\geq$. If below, use $<$ or $\leq$. (You can also test a point like $(0, 0)$ if it's not on the line.)

Which inequality is represented by the graph?

A) $y < -\dfrac{1}{2}x + 4$
B) $y > -\dfrac{1}{2}x + 4$
C) $y < -\dfrac{1}{2}x - 4$
D) $y < -\dfrac{1}{2}x + 8$

The boundary line has $y$-intercept $(0, 4)$ and $x$-intercept $(8, 0)$. Slope: $m = \dfrac{0 - 4}{8 - 0} = -\dfrac{1}{2}$. So the line is $y = -\dfrac{1}{2}x + 4$. This eliminates C and D.

The line is dashed (strict inequality), and shading is below the line, so $y < -\dfrac{1}{2}x + 4$. The answer is A.

Which inequality is represented by the graph?

A) $y < x + 4$
B) $y > -x + 4$
C) $y < -x + 4$
D) $y > x + 4$

The line passes through $(0, 4)$ and $(-4, 0)$. Slope: $m = \dfrac{4 - 0}{0 - (-4)} = 1$. Equation: $y = x + 4$. Dashed line with shading above, so $y > x + 4$. The answer is D.

 

Reading a Graph: Finding a Solution Point

If given a shaded-region graph and asked which point is a solution, look for the point that lies inside the shaded region (not on a dashed boundary).

Which ordered pair $(x, y)$ is a solution?

A) $(7, 0)$
B) $(5, -2)$
C) $(6, 6)$
D) $(2, 3)$

The graph shows a dashed vertical line at $x = 5$ with shading to the left ($x < 5$). Only $(2, 3)$ has $x < 5$. The answer is D.

If the boundary is a sloped line, you can test each option by substituting into the boundary equation. Points on the correct side of the line are solutions.

 

Key Strategies

For context interpretation questions, the two traps are (1) swapping the variables and (2) flipping the inequality direction. Read carefully which variable comes first in the ordered pair.

For graph questions, always start with the $y$-intercept to narrow your options. Then check slope sign (positive or negative) and shading direction. A quick test point like $(0, 0)$ can confirm the direction if the origin isn't on the boundary line.