Solve the Inequality for x -2< 8-2x< =-1

Solution:

Step 1: Isolate the variable $x$ in the inequality

• Begin by eliminating constants from the middle part of the inequality.

Step 1.1: Subtract $8$ from all parts of the inequality

• Since $8$ does not include $x$, we remove it by subtraction: $-2 - 8 < -2x \leq -1 - 8$

Step 1.2: Simplify the left section

• Perform the subtraction: $-10 < -2x \leq -1 - 8$

Step 1.3: Simplify the right section

• Complete the subtraction: $-10 < -2x \leq -9$

Step 2: Normalize the inequality by the coefficient of $x$

• Divide the entire inequality by $-2$, noting that the inequality signs will flip: $\frac{-10}{-2} > \frac{-2x}{-2} \geq \frac{-9}{-2}$

Step 3: Calculate the leftmost value

• Divide $-10$ by $-2$: $5 > \frac{-2x}{-2} \geq \frac{-9}{-2}$

Step 4: Simplify the variable terms

• Remove the common factor of $-2$ from the variable terms.

Step 4.1: Cancel out the $-2$ in the variable term

• Simplify the fraction: $5 > \frac{\cancel{-2} x}{\cancel{-2}} \geq \frac{-9}{-2}$

Step 4.2: Express $x$ as a single variable

• Since dividing by $1$ does not change the value: $5 > x \geq \frac{-9}{-2}$

Step 5: Recognize that dividing two negatives yields a positive

• Simplify to get the positive form: $5 > x \geq \frac{9}{2}$

Step 6: Arrange the interval correctly

• Ensure the smaller number is on the left: $\frac{9}{2} \leq x < 5$

Step 7: Present the solution in different formats

• The solution can be written as:

  Inequality Form: $\frac{9}{2} \leq x < 5$ Interval Notation: $\left[\frac{9}{2}, 5\right)$

Knowledge Notes:

• Inequalities: An inequality shows the relationship between two expressions that are not equal. When solving inequalities, similar rules to equations apply, but special attention is needed when multiplying or dividing by negative numbers, as this reverses the inequality sign.

• Interval Notation: This is a way of writing subsets of the real number line. An interval includes all numbers between two endpoints. For example, $[a, b)$ includes all numbers from $a$ to $b$, including $a$ but not $b$.

• Dividing Negative Numbers: When both the numerator and denominator of a fraction are negative, the result is a positive number because the negatives cancel out.

• Flipping Inequality Signs: When both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed to maintain the correct relationship.

• Simplifying Expressions: When solving for a variable, it's important to simplify expressions by canceling out common factors and performing arithmetic operations to isolate the variable.