Solving Linear Inequalities & Compound

[EXEC: MICRO_CORE]

✖️ 1. Applying equivalent transformations to inequalities (similar to equations)

⚖️ Transforming Inequalities Like Equations

You can add or subtract the same number from both sides without changing the inequality direction.
You can multiply or divide both sides by a positive number without flipping the sign.
The inequality sign stays the same for these operations: $<$ , $>$ , $\leq$ , $\geq$ .
Combine like terms and isolate the variable exactly like solving equations.
Always perform the same operation on both sides to maintain balance.

Example: Solve $x + 7 < 12$ . Subtract 7 from both sides: $x < 5$ .

💡 Think balance scale: what you do to one side, do to the other—sign stays put!

[EXEC: DEEP_COMPUTE]

1. Applying equivalent transformations to inequalities (similar to equations)

Equivalent Transformations for Inequalities

An equivalent transformation preserves the solution set of an inequality, just as it does for equations. We can add or subtract the same quantity from both sides, or multiply/divide both sides by a positive number without changing the inequality's direction.

Intuition: Think of inequality as a balance scale tilted to one side; adding equal weights to both sides keeps the tilt direction unchanged.

Core Rules:

If $a < b$ , then $a + c < b + c$ for any real $c$ .
If $a < b$ , then $a - c < b - c$ for any real $c$ .
If $a < b$ and $c > 0$ , then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$ .
Caution: Multiplying or dividing by a negative requires reversing the inequality sign (covered in the next block).

Consequence: These transformations allow systematic isolation of the variable, mirroring equation-solving techniques.

Example: Solve $2x + 5 < 13$ . Subtract 5: $2x < 8$ . Divide by 2: $x < 4$ .

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["solving_linear_block_1"] | DIFF_LEVEL:

[EXEC: MICRO_CORE]

✖️ 2. The reversal rule: logically justifying why we flip the inequality sign when multiplying/dividing by negatives

🔄 Flipping the Sign When Multiplying by Negatives

When you multiply or divide both sides by a negative number, you must flip the inequality sign.
This happens because negatives reverse the order on the number line.
If $a < b$ , then $-a > -b$ (the smaller becomes larger when negated).
Forgetting to flip is the most common mistake in inequality solving.
Only multiplication/division by negatives triggers the flip—not addition or subtraction.

Example: Solve $-3x > 9$ . Divide both sides by $-3$ and flip: $x < -3$ .

💡 Negative flips the script: imagine the number line mirrored around zero!

[EXEC: DEEP_COMPUTE]

2. The reversal rule: logically justifying why we flip the inequality sign when multiplying/dividing by negatives

The Reversal Rule for Negative Multipliers

Definition: When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed to preserve truth. This is the reversal rule.

Intuition: Multiplying by a negative reflects the number line across zero, flipping the order. If $3 < 5$ , then $-3 > -5$ because $-3$ lies to the right of $-5$ on the number line.

Core Rules:

If $a < b$ and $c < 0$ , then $ac > bc$ .
If $a < b$ and $c < 0$ , then $\frac{a}{c} > \frac{b}{c}$ .
Failure to reverse produces incorrect solution sets.
This rule applies to all inequality symbols: $<, \leq, >, \geq$ .

Consequence: The reversal rule is non-negotiable; omitting it invalidates the solution.

Example: Solve $-3x \leq 9$ . Divide by $-3$ and reverse: $x \geq -3$ .

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["solving_linear_block_2"] | DIFF_LEVEL:

[EXEC: MICRO_CORE]

✖️ 3. Solving compound inequalities with AND (intersections) and OR (unions)

🔗 Compound Inequalities: AND vs OR

AND means both conditions must be true simultaneously (intersection of solutions).
OR means at least one condition must be true (union of solutions).
For AND: solve each inequality separately, then find the overlap on the number line.
For OR: solve each inequality separately, then combine all solutions.
Write AND as $a < x < b$ when possible; OR stays as two separate inequalities.

Example: Solve $2 < x + 1 < 5$ . Subtract 1 from all parts: $1 < x < 4$ (AND gives the interval between 1 and 4).

💡 AND = overlap zone; OR = everything covered by either condition!

[EXEC: DEEP_COMPUTE]

3. Solving compound inequalities with AND (intersections) and OR (unions)

Compound Inequalities: Intersections and Unions

Definition: A compound inequality combines two inequalities using logical connectors. An AND compound (intersection) requires both conditions to hold simultaneously; an OR compound (union) requires at least one condition to hold.

Intuition: AND narrows the solution to the overlap region; OR broadens it to include all solutions satisfying either inequality.

Core Rules:

AND: $a < x$ AND $x < b$ is written $a < x < b$ ; solution is the intersection $[a, b]$ (or $(a, b)$ if strict).
OR: $x < a$ OR $x > b$ yields two disjoint intervals: $(-\infty, a) \cup (b, \infty)$ .
Solve each inequality separately, then combine results according to the connector.
Graph solutions on a number line to visualize intersections or unions.

Consequence: Misidentifying the connector leads to incorrect solution sets.

Example: Solve $-2 \leq 3x + 1 < 7$ . Subtract 1: $-3 \leq 3x < 6$ . Divide by 3: $-1 \leq x < 2$ .

WARN: PRACTICE_BLOCK_EMPTY

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[EXEC: MICRO_CORE]

✖️ 4. Applications: Calculating budget constraints in finance and maximum payload capacities in structural engineering

💰 Real-World Constraints: Budgets and Payloads

Budget constraints: Total spending must stay below or equal to available funds.
Set up inequality: cost per item times quantity $\leq$ total budget.
Payload capacity: Total weight must not exceed structural limits.
Set up inequality: weight per unit times quantity $\leq$ maximum capacity.
Solve for the variable to find the maximum allowable quantity.

Example: A truck holds max 2000 kg. Each box weighs 150 kg. Solve $150x \leq 2000$ : $x \leq 13.33$ , so max 13 boxes.

💡 Inequality = safety margin: never exceed the limit!

[EXEC: DEEP_COMPUTE]

4. Applications: Calculating budget constraints in finance and maximum payload capacities in structural engineering

Applications in Finance and Engineering

Definition: Linear inequalities model real-world constraints where a quantity must remain within specified bounds. Budget constraints limit total expenditure; payload capacities restrict maximum load.

Intuition: Inequalities express feasibility conditions—what is permissible rather than exact.

Core Rules:

Budget constraint: If total cost is $C = ax + by$ and budget is $B$ , then $ax + by \leq B$ .
Payload capacity: If total weight is $W = m_1 + m_2 + \cdots$ and max capacity is $W_{\text{max}}$ , then $W \leq W_{\text{max}}$ .
Solve for the variable of interest to find feasible ranges.
Interpret solutions in context (e.g., non-negative quantities only).

Consequence: Violating these inequalities results in budget overruns or structural failure.

Example: A truck has max capacity 5000 kg. If it carries 1200 kg of steel and $x$ kg of gravel, then $1200 + x \leq 5000$ , so $x \leq 3800$ kg.

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["solving_linear_block_4"] | DIFF_LEVEL:

Solving linear inequalities

✖️ 1. Applying equivalent transformations to inequalities (similar to equations)

⚖️ Transforming Inequalities Like Equations

1. Applying equivalent transformations to inequalities (similar to equations)

Equivalent Transformations for Inequalities

WARN: PRACTICE_BLOCK_EMPTY

✖️ 2. The reversal rule: logically justifying why we flip the inequality sign when multiplying/dividing by negatives

🔄 Flipping the Sign When Multiplying by Negatives

2. The reversal rule: logically justifying why we flip the inequality sign when multiplying/dividing by negatives

The Reversal Rule for Negative Multipliers

WARN: PRACTICE_BLOCK_EMPTY

✖️ 3. Solving compound inequalities with AND (intersections) and OR (unions)

🔗 Compound Inequalities: AND vs OR

3. Solving compound inequalities with AND (intersections) and OR (unions)

Compound Inequalities: Intersections and Unions

WARN: PRACTICE_BLOCK_EMPTY

✖️ 4. Applications: Calculating budget constraints in finance and maximum payload capacities in structural engineering

💰 Real-World Constraints: Budgets and Payloads

4. Applications: Calculating budget constraints in finance and maximum payload capacities in structural engineering

Applications in Finance and Engineering

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