Inequalities: Step-by-Step Solutions

Question

I'm really struggling to get my head around inequalities for my math class. I keep getting the steps mixed up, especially with multiplying/dividing by negative numbers. Can someone show me how to solve them, one step at a time?

CharlesMiller11 · Accepted Answer

Understanding Inequalities ➕➖

Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities involves finding the range of values that satisfy the given inequality.

Basic Principles 📜

Here are some fundamental principles for solving inequalities:

Addition/Subtraction Property: Adding or subtracting the same number from both sides of an inequality does not change the inequality.
Multiplication/Division Property: Multiplying or dividing both sides by the same positive number does not change the inequality. However, multiplying or dividing by a negative number reverses the inequality sign.
Transitive Property: If $a < b$ and $b < c$, then $a < c$.

Solving Linear Inequalities ✏️

Let's solve the linear inequality $3x + 5 < 14$:

Isolate the variable term: Subtract 5 from both sides:
```
3x + 5 - 5 < 14 - 5
3x < 9
```
Solve for x: Divide both sides by 3:
```
3x / 3 < 9 / 3
x < 3
```
Solution: The solution is $x < 3$, meaning any value of x less than 3 will satisfy the original inequality.

Solving Compound Inequalities 🧭

Compound inequalities involve two or more inequalities combined. For example, $ -3 < 2x + 1 ≤ 7 $.

Isolate the variable term: Subtract 1 from all parts:
```
-3 - 1 < 2x + 1 - 1 ≤ 7 - 1
-4 < 2x ≤ 6
```
Solve for x: Divide all parts by 2:
```
-4 / 2 < 2x / 2 ≤ 6 / 2
-2 < x ≤ 3
```
Solution: The solution is $-2 < x ≤ 3$.

Solving Absolute Value Inequalities 💡

Absolute value inequalities involve absolute value expressions. For example, $|x - 2| < 3$.

Rewrite as a compound inequality:
```
-3 < x - 2 < 3
```
Solve for x: Add 2 to all parts:
```
-3 + 2 < x - 2 + 2 < 3 + 2
-1 < x < 5
```
Solution: The solution is $-1 < x < 5$.

Multiplying or Dividing by a Negative Number ⛔

Consider the inequality $-2x > 6$. To solve for $x$, we must divide by $-2$. Remember to reverse the inequality sign:

-2x / -2 < 6 / -2
x < -3

Therefore, the solution is $x < -3$.

Graphing Inequalities 📈

Solutions to inequalities can be graphically represented on a number line. For $x < 3$, draw a number line and shade the region to the left of 3. Use an open circle at 3 to indicate that 3 is not included in the solution.