Solving a Less Than Inequality

Solve $|x - 2| < 5$

Recognize the type: $|x - 2| < 5$ is a "less than" inequality = Expression is within 5 units of zero

Write as compound inequality: For $|A| < a$, write $-a < A < a$ = $-5 < x - 2 < 5$

Solve by adding 2 to all parts: $-5 + 2 < x - 2 + 2 < 5 + 2$ = $-3 < x < 7$

Write solution set: All numbers between $-3$ and $7$ = $x \in (-3, 7)$

Answer: $-3 < x < 7$ or in interval notation: $(-3, 7)$

Solving a Greater Than Inequality

Solve $|2x + 1| > 7$

Recognize the type: $|2x + 1| > 7$ is a "greater than" inequality = Expression is more than 7 units from zero

Write as two separate inequalities: For $|A| > a$, write $A < -a$ OR $A > a$ = $2x + 1 < -7$ OR $2x + 1 > 7$

Solve the first inequality: $2x + 1 < -7 \Rightarrow 2x < -8 \Rightarrow x < -4$ = $x < -4$

Solve the second inequality: $2x + 1 > 7 \Rightarrow 2x > 6 \Rightarrow x > 3$ = $x > 3$

Combine with OR: Solution includes both regions = $x < -4$ or $x > 3$

Answer: $x < -4$ or $x > 3$, which is $(-\infty, -4) \cup (3, \infty)$

Inequality with Coefficient

Solve $|3x - 6| \leq 9$

Identify type and setup: "Less than or equal" type: write as compound inequality = $-9 \leq 3x - 6 \leq 9$

Add 6 to all parts: $-9 + 6 \leq 3x - 6 + 6 \leq 9 + 6$ = $-3 \leq 3x \leq 15$

Divide all parts by 3: $\frac{-3}{3} \leq \frac{3x}{3} \leq \frac{15}{3}$ = $-1 \leq x \leq 5$

Write in interval notation: Include endpoints (closed brackets) = $x \in [-1, 5]$

Answer: $-1 \leq x \leq 5$ or $[-1, 5]$

Special Case: No Solution

Solve $|x + 4| < -2$

Consider what absolute value means: Absolute value is always $\geq 0$ = $|x + 4| \geq 0$ for all $x$

Compare to the inequality: We need $|x + 4| < -2$, but absolute value cannot be negative = No value can satisfy this

State the conclusion: It's impossible for a non-negative number to be less than $-2$ = No solution

Answer: No solution (the solution set is empty: $\emptyset$)

Special Case: All Real Numbers

Solve $|x - 1| > -5$

Consider what absolute value means: Absolute value is always $\geq 0$ = $|x - 1| \geq 0$ for all $x$

Compare to the inequality: Any non-negative number is greater than $-5$ = Every $x$ works

State the conclusion: Since $|x - 1| \geq 0 > -5$ always = All real numbers

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

Mistake: Writing $|x| < 3$ as $x < 3$ or $x < -3$

Why: Students forget that "less than" creates an AND compound inequality (between two values), not an OR.

Correct: $|x| < 3$ means $-3 < x < 3$ (values between $-3$ and $3$)

Mistake: Writing $|x| > 3$ as $-3 < x < 3$

Why: Confusing the two cases. "Greater than" means the value is FAR from zero, not close to it.

Correct: $|x| > 3$ means $x < -3$ OR $x > 3$ (values outside the interval)

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

Why: While solving, if you multiply or divide by a negative number, the inequality direction reverses.

Correct: Be careful when the coefficient of $x$ is negative inside the absolute value

Mistake: Not checking if the inequality has no solution or all real numbers

Why: Absolute value is always non-negative, so $|A| < \text{negative}$ has no solution, and $|A| > \text{negative}$ is always true.

Correct: Always check: Can this inequality actually be satisfied?

Manufacturing Tolerances

Factory machines must produce parts within specific tolerances. A bolt must be 10 mm with tolerance of 0.05 mm.

The inequality $|d - 10| \leq 0.05$ means the diameter $d$ must be between 9.95 mm and 10.05 mm.

Temperature Control

Many products require storage within specific temperature ranges.

Vaccines must be stored at 4 degrees Celsius with variation no more than 2 degrees: $|T - 4| \leq 2$, so $2 \leq T \leq 6$ degrees.

Speed Limits and Radar

Police radar has a margin of error. A driver might contest a ticket if the measured speed is within the error margin.

If radar accuracy is 3 km/h and the limit is 50 km/h, speeds registering between 47 and 53 might be within error: $|v - 50| \leq 3$.

Q: How do I remember when to use AND vs OR?

A: Think of distance: "less than" means CLOSE to zero (between two values = AND). "Greater than" means FAR from zero (outside = OR). Memory trick: Less than = beLEss = betweenLESS = between.

Q: What if the absolute value equals a number, not less than or greater than?

A: For $|A| = a$, you get exactly two solutions: $A = a$ or $A = -a$. This is an equation, not an inequality.

Q: Can the number on the right side be negative?

A: Yes, but check carefully! Since absolute value is always non-negative: $|A| < \text{negative}$ has no solution, while $|A| > \text{negative}$ is true for all values.