Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each value of \(x\) is a solution of the inequality. Inequality $$-1<\frac{3-x}{2} \leq 1$$ Values (a) \(x=-1\) (b) \(x=\sqrt{5}\) (c) \(x=1\) (d) \(x=5\)

Step-by-Step Solution

Answer

The values \(x=-1\) and \(x=1\) are solutions to the inequality \(-1<\frac{3-x}{2} \leq 1\), while the values \(x=\sqrt{5}\) and \(x=5\) are not solutions.

Step 1 Substitute the value (a)

First substitute \(x=-1\) into the inequality \(-1<\frac{3-x}{2} \leq 1\). This results in: \(-1<\frac{3+1}{2}\) which simplifies to \(-1<2\) which is a true statement. Hence, \(x=-1\) is a solution to the inequality.

Step 2 Substitute the value (b)

Meanwhile, substitute \(x=\sqrt{5}\) into the inequality: \(-1<\frac{3-\sqrt{5}}{2}\). This simplifies to: \(-1<-\frac{\sqrt{5}-3}{2}\) which is false. Therefore, \(x=\sqrt{5}\) is not a solution of the inequality.

Step 3 Substitute the value (c)

Substitute \(x=1\) into the inequality: \(-1<\frac{3-1}{2}\). This simplifies to: \(-1<1\) which is a true statement. Thus, \(x=1\) is a solution to the inequality.

Step 4 Substitute the value (d)

Substitute \(x=5\) into the inequality: \(-1<\frac{3-5}{2}\). This simplifies to: \(-1<-1\), which is false. Hence, \(x=5\) is not a solution to the inequality.

Key Concepts

Solving Inequalities

Solving inequalities involves finding the set of values that satisfy a particular condition. An inequality, unlike an equation, contains signs such as \( <, >, \leq, \) and \( \geq \), which mean "less than", "greater than", "less than or equal to", and "greater than or equal to" respectively. When solving inequalities, the ultimate goal is to establish which values make the inequality true. To solve these, you often need to perform similar operations as you would with equations. However, remember that if you multiply or divide the inequality by a negative number, the inequality sign must be reversed. For example, if you have an inequality like \(-2x > 6\), you would divide both sides by \(-2\), which yields \(x < -3\). The key is ensuring the direction of the inequality remains correct once those operations are done.

Substitution Method

The substitution method is a technique that allows you to determine if specific values satisfy an equation or inequality. It is very simple to apply. Here's how it works:

• Take the value you want to test.
• Replace the variable in the inequality or equation with this value.
• Simplify the expression to see if it holds true.

In our case with the inequality \(-1 < \frac{3-x}{2} \leq 1\), let's say you substitute \(x = 1\). You will replace \(x\) with 1, calculate the expression, and check whether the inequality is satisfied. If the inequality holds true, then the substituted value is a solution.This method is particularly helpful when checking multiple values, one by one, like when given a list from which we need to identify which ones hold true.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. A fundamental understanding of these expressions is essential when solving inequalities, as they often form the core component of such problems.For the expression \(\frac{3-x}{2}\), you have the variable \(x\) and constants both in the numerator (3) and in the denominator (2). This expression simplifies differently based on the value substituted for \(x\). Manipulating algebraic expressions requires several skills:

• Identifying which operations to perform first (following the order of operations).
• Simplifying fractions to their lowest terms when necessary.
• Rewriting expressions to isolate the variable on one side.

Understanding these concepts will make it easier to interpret and solve related inequalities with accuracy.

Step-by-Step Solutions

Providing a step-by-step solution is crucial in understanding how to approach and solve algebraic inequalities effectively. This methodical approach helps in breaking down the problem into manageable parts, allowing for thorough comprehension.In the original problem:

• Each solution starts by substituting the given \(x\) value into the inequality.
• The inequality is simplified to verify its truth.
• A decision is made whether the substituted value is a solution.

Every step corresponds to an essential part of problem-solving. For instance, substituting the value \(x = -1\) into the inequality \(-1 < \frac{3-x}{2} \leq 1\) shows how to handle the expression algebraically to decide if it holds true. Remember, these steps do more than just provide answers. They teach the strategy behind algebraic problem-solving, allowing you to tackle similar challenges independently in the future.