Solve Linear Inequalities: One Variable 3 | Effective Methods in Algebra 2

What are Linear Inequalities in One Variable?
Linear inequalities in one variable are mathematical expressions involving inequality symbols (`<`, `>`, `?`, `?`) and a linear expression in a single variable. These inequalities describe a range of values that satisfy the inequality condition. For example, `2x + 3 < 7` is a linear inequality.

How Do We Solve Linear Inequalities in One Variable?
Solving linear inequalities involves isolating the variable on one side while maintaining the inequality’s balance. Here is a step-by-step guide to solving such inequalities:

Step 1: Simplify Both Sides
Simplify both sides of the inequality if needed by combining like terms and ensuring all terms involving the variable are on one side and constants on the other.

Question: How do we simplify inequalities?
Suppose we have the inequality `2x + 3 < 7`. Subtract 3 from both sides to isolate the term with the variable.
```
2x + 3 - 3 < 7 - 3
2x < 4
```

Step 2: Isolate the Variable
Isolate the variable by performing basic algebraic operations such as addition, subtraction, multiplication, or division.

Question: How do we isolate the variable when we have `2x < 4`?
Here, divide both sides by 2 to solve for x.
```
2x / 2 < 4 / 2
x < 2
```

Step 3: Reverse the Inequality Symbol if Multiplying or Dividing by a Negative Number
When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.

Question: How does multiplying or dividing by a negative number affect the inequality?
Example: Solve `-3x > 6`.
First, divide both sides by `-3` and reverse the inequality symbol:
```
-3x / -3 < 6 / -3
x < -2
```

Step 4: Represent the Solution
The solution to the inequality can be represented on a number line or using interval notation.

Question: How do we represent `x < 2` on a number line and using interval notation?
1. On a number line: Draw a number line and shade all values to the left of 2. Use an open circle at 2 to indicate that 2 is not included in the solution.
2. Interval notation: Express this as `(-?, 2)`.

Additional Example
Question: How do we solve and represent the inequality `5 - 2x ? 1`?
1. Simplify: Subtract 5 from both sides.
```
5 - 2x - 5 ? 1 - 5
-2x ? -4
```
2. Isolate the variable by dividing both sides by `-2` and reverse the inequality symbol:
```
-2x / -2 ? -4 / -2
x ? 2
```
3. Represent the solution:
- On a number line: Draw a number line and shade all values to the left of 2, including 2, using a closed circle at 2.
- Interval notation: Express this as `(-?, 2]`.

Key Takeaways:
- When solving linear inequalities, simplify both sides and isolate the variable.
- Be mindful of reversing the inequality symbol when multiplying or dividing by a negative number.
- Represent solutions using number lines and interval notation for clarity.

Algebra 2: Solve Linear Inequalities: One Variable 3 | Effective Methods

Related

Recommended Videos