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Inequality substitution is a method where specific values are plugged into an inequality to test if they satisfy the condition. This technique is particularly useful when the range of potential solutions is discrete or limited.

For instance, consider the inequality \[\begin{equation} -1<\frac{3-x}{2} \leq 1\end{equation}\]. To find out if a particular value of \(x\) is a solution, we replace \(x\) with that value and simplify the expression to see if the inequality holds true.

Example: When \(x=0\), the substitution gives us \[\begin{equation} -1<\frac{3}{2} \leq 1\end{equation}\] which simplifies to \[\begin{equation} -1<1.5 \leq 1\end{equation}\]. This tells us that \(x=0\) is not a solution since 1.5 is not less than or equal to 1.

Algebraic reasoning involves using mathematical logic and properties to solve for variables within equations and inequalities. This type of reasoning is the backbone of algebra and helps us understand the relationships between variables and constants.

In the context of the inequality \[\begin{equation} -1<\frac{3-x}{2} \leq 1\end{equation}\], algebraic reasoning guides the approach to equate or compare the resulting expression after substitution with the limits of the inequality. When a value of \(x\) maintains the inequality's truth, it is considered a solution.

Important Point: It's crucial to remember the rules for inequalities when performing operations on them, such as the fact that multiplying or dividing by a negative number reverses the inequality sign.

Inequality solutions refer to the set of all possible values of the variable that satisfy the inequality. Solutions to inequalities can sometimes be a range of values or specific discrete numbers, which can be represented on a number line or using interval notation.

For the given exercise, we are searching for discrete solutions that fulfill the condition of the inequality \[\begin{equation} -1<\frac{3-x}{2} \leq 1\end{equation}\]. If the substituted value keeps the inequality true after we perform the necessary arithmetic, then it is part of the solution set. From the exercise, we find that only \(x=1\) is a solution as it makes the inequality valid (e.g., -1<1 \leq 1).

Rational expressions are fractions that contain polynomials in the numerator, denominator, or both. These expressions appear frequently in algebra, and working with them often requires simplifying or finding common denominators.

In the exercise, the inequality \[\begin{equation} -1<\frac{3-x}{2} \leq 1\end{equation}\], involves a rational expression where the numerator is an expression \[\begin{equation}(3-x)\end{equation}\] and the denominator is 2. When evaluating whether a value of \(x\) satisfies the inequality, it's important to simplify the rational expression correctly and take note of any restrictions, such as values that may cause division by zero, which do not apply in this particular case.