Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

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> [!TLDR] > We need consecutive odd positive integers less than 10 whose sum exceeds 11. The pairs are **(5, 7)** and **(7, 9)**. ## Why This Problem Is About Linear Inequalities The question asks for *all pairs* satisfying two conditions: each integer is less than 10, and their sum is more than 11. This is a classic linear inequality problem in two variables, but because the integers are consecutive and odd, we can reduce it to a single variable. Let the smaller odd integer be $x$. Since they are consecutive odd numbers, the next one is $x + 2$. Both are positive and less than 10, so: $$ x > 0 \quad ext{and} \quad x + 2 < 10 $$ The sum condition gives: $$ x + (x + 2) > 11 $$ We now solve these inequalities together. ## Step-by-Step Solution **1. Set up the variable and constraints** Let the smaller odd positive integer be $x$. Then the larger is $x + 2$. Both are positive: $x > 0$. Both are smaller than 10: $x < 10$ and $x + 2 < 10$ — the second is stricter, so we use $x < 8$. > [!NOTE] > Since $x$ and $x+2$ are both less than 10, the condition $x+2 < 10$ automatically ensures $x < 10$. So the upper bound is $x < 8$. **2. Write the sum inequality** The sum is more than 11: $$ x + (x + 2) > 11 $$ Simplify: $$ 2x + 2 > 11 $$ Subtract 2 from both sides: $$ 2x > 9 $$ Divide by 2: $$ x > 4.5 $$ Since $x$ is an odd positive integer, $x$ must be at least 5. **3. Combine the inequalities** We have: $$ x > 4.5 \quad ext{and} \quad x < 8 $$ Also $x$ is odd and positive. So the possible integer values for $x$ are: $$ x = 5, 7 $$ …

Symbol	Meaning	Example	Reads as
$<$	less than	$x < 5$	x is less than 5
$>$	greater than	$x > 5$	x is greater than 5
$\le$	less than or equal to	$x \le 5$	x is at most 5
$\ge$	greater than or equal to	$x \ge 5$	x is at least 5

	Carbs	Protein	Fat	Fibre
Food-1	20	5	3	2
Food-2	10	2	4	5

Nutrient	Constraint
Carbs (min 45)	$20x + 10y \ge 45$
Protein (min 25)	$5x + 2y \ge 25$
Fat (min 15)	$3x + 4y \ge 15$
Fibre (min 15)	$2x + 5y \ge 15$

Q.Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

Concept understanding — Linear Inequalities

Linear Inequalities: The Intuition First

The Four Symbols

Solving Linear Inequalities: Almost Like Equations

The Golden Rule (and the only trap)

Example: Solve $2x + 3 < 7$

Example: Solve $-3x + 5 \ge 11$

Representing Solutions: The Number Line

The Precise Definition

Why This Problem Is About Linear Inequalities

Step-by-Step Solution

Step 1 — clear denominators (LCM = 6)

Step 2 — expand and collect

Step 3 — isolate $x$

Step 1 — Define the variables

Step 2 — Write the output

Step 3 — Translate the constraints

Step 1 — Express all sides via the shortest side $s$

Step 2 — Write the perimeter inequality

Step 3 — Solve the inequality

Step 1 — Remove the fractions

Step 2 — Isolate $x$

Step 3 — Write the solution set

Step 1 — Interpret "at least"

Step 2 — Form one constraint per nutrient

Step 3 — Add non-negativity

Q.Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

Why This Problem Is About Linear Inequalities

Step-by-Step Solution

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