Solve the Inequality for d -2< =(2d-2)/4< 3

Solution:

Simplify the Inequality

Step 1: Simplify the fraction

• Begin by factoring out the common factor of $2$ from the numerator of the fraction.

  $- 2 \leq \frac{2(d - 1)}{4} < 3$

• Then, reduce the fraction by canceling out the common factor of $2$ in the numerator and denominator.

  $- 2 \leq \frac{d - 1}{2} < 3$

Step 2: Eliminate the denominator

• Multiply all parts of the inequality by $2$ to remove the denominator.

  $- 4 \leq d - 1 < 6$

Step 3: Solve for $d$

• Isolate the variable $d$ by adding $1$ to all parts of the inequality.

  $- 3 \leq d < 7$

Step 4: Express the solution

• The solution can be expressed as:

  Inequality Form: $- 3 \leq d < 7$ Interval Notation: $[-3, 7)$

Knowledge Notes:

To solve the given inequality, $d - 2 \leq (2d - 2)/4 < 3$, we need to apply several algebraic techniques:

1. Factoring: This involves taking out a common factor from terms to simplify expressions. In this case, we factor out a $2$ from the numerator of the fraction.

2. Reducing Fractions: After factoring, we can cancel out common factors in the numerator and denominator to simplify the fraction.

3. Multiplication Principle: We can multiply or divide all parts of an inequality by the same positive number without changing the direction of the inequality. This helps to eliminate fractions.

4. Isolation of the Variable: To solve for the variable, we need to get it by itself on one side of the inequality. This often involves adding or subtracting terms from all parts of the inequality.

5. Interval Notation: This is a way of writing the set of solutions to an inequality. For the inequality $a \leq x < b$, the interval notation is $[a, b)$, which includes $a$ but not $b$.

By following these steps, we can solve the inequality and express the solution in both inequality form and interval notation.