1. Q Essential Question How is solving a two-step inequality similar to and different from solving a two-step equation?

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To understand how solving a two-step inequality is similar to and different from solving a two-step equation, let's break down the concepts step by step.

### Similarities

1. **Structure**: Both two-step inequalities and two-step equations involve two operations to isolate the variable. Typically, these operations include addition or subtraction followed by multiplication or division.

2. **Goal**: The goal in both cases is to solve for the variable, whether it be in an equation or an inequality.

3. **Order of Operations**: In both cases, the order of operations (PEMDAS/BODMAS) applies. You perform operations in a specific order to isolate the variable.

### Differences

1. **Type of Solution**:
   - **Equation**: A two-step equation results in a specific value for the variable (e.g., $$ x = 5 $$).
   - **Inequality**: A two-step inequality results in a range of values (e.g., $$ x < 5 $$ or $$ x \geq 5 $$).

2. **Direction of the Inequality**: 
   - When solving inequalities, if you multiply or divide both sides by a negative number, you must reverse the inequality sign (e.g., if $$ -2x > 4 $$, dividing by -2 gives $$ x < -2 $$).
   - This reversal does not occur in equations.

3. **Graphical Representation**:
   - The solution to an equation can be represented as a single point on a number line.
   - The solution to an inequality is represented as a range of values, often shown with a shaded region on a number line.

### Example

Let's illustrate both concepts with an example:

**Two-Step Equation**:
1. Solve $$ 2x + 3 = 7 $$.
   - Subtract 3 from both sides: $$ 2x = 4 $$.
   - Divide by 2: $$ x = 2 $$.

**Two-Step Inequality**:
1. Solve $$ 2x + 3 < 7 $$.
   - Subtract 3 from both sides: $$ 2x < 4 $$.
   - Divide by 2: $$ x < 2 $$.

### Conclusion

In summary, while both two-step equations and inequalities share a similar structure and goal, they differ significantly in the nature of their solutions, the treatment of operations (especially with negative numbers), and their graphical representations.
Solving a two-step inequality is similar to solving a two-step equation in that both involve two operations to isolate the variable and follow the order of operations. However, the key differences are that an inequality results in a range of values rather than a single value, and when multiplying or dividing by a negative number, the inequality sign must be reversed.

1. Q Essential Question How is solving a two-step inequality similar to and different from solving a two-step equation?

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