Solving an AND Inequality

Solve: $-3 < 2x + 1 < 7$

Understand the compound inequality: This means $-3 < 2x + 1$ AND $2x + 1 < 7$ simultaneously = Both must be true

Subtract 1 from all three parts: $-3 - 1 < 2x + 1 - 1 < 7 - 1$ = $-4 < 2x < 6$

Divide all three parts by 2: $\frac{-4}{2} < \frac{2x}{2} < \frac{6}{2}$ = $-2 < x < 3$

Write in interval notation: Open circles at $-2$ and $3$ (not included) = $(-2, 3)$

Answer: $-2 < x < 3$ or in interval notation: $(-2, 3)$

Solving an OR Inequality

Solve: $3x - 2 < -8$ OR $3x - 2 > 4$

Solve the first inequality: $3x - 2 < -8 \Rightarrow 3x < -6 \Rightarrow x < -2$ = $x < -2$

Solve the second inequality: $3x - 2 > 4 \Rightarrow 3x > 6 \Rightarrow x > 2$ = $x > 2$

Combine with OR: The solution includes values in either region = $x < -2$ OR $x > 2$

Write in interval notation: Union of two intervals = $(-\infty, -2) \cup (2, \infty)$

Answer: $x < -2$ OR $x > 2$, or in interval notation: $(-\infty, -2) \cup (2, \infty)$

AND Inequality with No Solution

Solve: $x > 5$ AND $x < 2$

Analyze the first condition: $x > 5$ means $x$ must be greater than 5 = Values: 6, 7, 8, ...

Analyze the second condition: $x < 2$ means $x$ must be less than 2 = Values: 1, 0, -1, ...

Find the intersection: No number can be both greater than 5 AND less than 2 = No overlap

State the solution: There is no value that satisfies both conditions = No solution: $\emptyset$

Answer: No solution ($\emptyset$) - the conditions are contradictory

OR Inequality with All Real Numbers

Solve: $x \leq 4$ OR $x \geq 1$

Analyze the first condition: $x \leq 4$ includes all numbers up to and including 4 = $(-\infty, 4]$

Analyze the second condition: $x \geq 1$ includes all numbers from 1 onwards = $[1, \infty)$

Find the union: Every real number satisfies at least one condition = Complete overlap

State the solution: Any number is either $\leq 4$ or $\geq 1$ (or both) = All real numbers: $\mathbb{R}$

Answer: All real numbers ($\mathbb{R}$) or $(-\infty, \infty)$

Real-World AND Inequality

A package must weigh at least 2 kg but no more than 10 kg to qualify for standard shipping. Write and solve an inequality for the weight $w$.

Identify the conditions: At least 2 kg: $w \geq 2$. No more than 10 kg: $w \leq 10$ = Two conditions

Combine with AND: Both conditions must be met simultaneously = $w \geq 2$ AND $w \leq 10$

Write as compound inequality: Combined into one statement = $2 \leq w \leq 10$

Write in interval notation: Closed brackets (endpoints included) = $[2, 10]$

Answer: $2 \leq w \leq 10$ or $[2, 10]$ kg

Mistake: Confusing AND with OR when graphing

Why: AND requires the overlap (intersection), while OR includes everything from both sets (union).

Correct: AND = intersection (smaller region, both conditions). OR = union (larger region, either condition).

Mistake: Forgetting to flip the inequality when multiplying/dividing by a negative

Why: When solving each part of a compound inequality, the same rules apply as with regular inequalities.

Correct: If you multiply or divide by a negative number, reverse ALL inequality signs in that step.

Mistake: Writing $5 < x < 2$ and thinking it has solutions

Why: This would require $x$ to be greater than 5 AND less than 2 simultaneously, which is impossible.

Correct: Always check if an AND inequality makes logical sense. If $a > b$ in $a < x < b$, there is no solution.

Mistake: Using the wrong interval notation brackets

Why: Square brackets $[$ $]$ mean the endpoint is included ($\leq$ or $\geq$), parentheses $($ $)$ mean it's excluded ($<$ or $>$).

Correct: $x > 3$: use $(3, \infty)$. $x \geq 3$: use $[3, \infty)$. Infinity always uses parentheses.

Healthy Heart Rate Zones

Fitness trainers use compound inequalities to define heart rate zones during exercise.

The fat-burning zone is when your heart rate is between 60% and 70% of your maximum. For a max of 190 bpm: $114 \leq h \leq 133$

Manufacturing Quality Control

Products must meet specifications within tolerance ranges. Items outside the range are rejected.

A bolt must be between 9.8 mm and 10.2 mm to pass inspection: $9.8 \leq d \leq 10.2$

Safe Driving Speed

Traffic laws use compound inequalities to define legal speed ranges.

On a highway, you must drive at least 60 km/h but no more than 130 km/h: $60 \leq s \leq 130$

Q: What's the difference between 'and' and 'or' in compound inequalities?

A: AND means both conditions must be true simultaneously (intersection). OR means at least one condition must be true (union). AND typically gives a smaller solution set; OR gives a larger one.

Q: How do I know when to use brackets vs parentheses in interval notation?

A: Use square brackets $[ ]$ when the endpoint is included ($\leq$ or $\geq$). Use parentheses $( )$ when the endpoint is NOT included ($<$ or $>$). Infinity ($\infty$) ALWAYS uses parentheses because you can never reach infinity.

Q: Can a compound inequality have no solution?

A: Yes! An AND inequality has no solution when the conditions contradict each other (like $x > 5$ AND $x < 2$). However, an OR inequality will always have a solution unless the individual inequalities themselves have no solutions.