Problem 13 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(3-\frac{1}{2} x>0\) (a) \(x=10\) (b) \(x=6\) (c) \(x=-\frac{3}{4}\) (d) \(x=0\)

Step-by-Step Solution

Answer

For \(x=10\) and \(x=6\), the inequality is false, and for \(x=-\frac{3}{4}\) and \(x=0\) the inequality is true.

Step 1: Check value (a)

Substitute \(x=10\) into the inequality. This becomes \(3-\frac{1}{2} \cdot 10\). After performing the multiplication and subtraction, this simplifies to \(-2 > 0\), which is false.

Step 2: Check value (b)

Substitute \(x=6\) into the inequality. This becomes \(3-\frac{1}{2} \cdot 6\). After performing the multiplication and subtraction, this simplifies to \(-0 > 0\), which is also false.

Step 3: Check value (c)

Substitute \(x=-\frac{3}{4}\) into the inequality. This becomes \(3-\frac{1}{2} \cdot -\frac{3}{4}\). After performing the multiplication and subtraction, this simplifies to \(3+\frac{3}{8}\), which equals \(\frac{27}{8} > 0\). This is true.

Step 4: Check value (d)

Substitute \(x=0\) into the inequality. This becomes \(3- \frac{1}{2} \cdot 0\), which simplifies to \(3>0\). This is true.

Key Concepts

Understanding Algebra in Inequality Solutions

When dealing with inequalities, algebra plays an essential role. Algebra allows us to manipulate equations and expressions in order to find unknowns or to simplify the process of solving problems. In the context of inequalities like the one in our original exercise, algebraic principles help us to rearrange terms and make the inequality simpler to fathom.

By breaking down expressions and applying fundamental rules such as the distributive law, we can compute more complex operations. Here, substituting values into a basic algebraic inequality requires us to be familiar with the rules of operations including multiplication, division, and addition within an equation.

Understanding basic algebra is crucial as it provides the foundational steps necessary to engage with more sophisticated mathematical inquiries, ensuring that you can plug in different values confidently and get accurate results.

Exploring Inequalities: Basic Concepts

Inequalities express the relation between two expressions that are not equal. They involve symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). In mathematical reasoning, inequalities help us determine the range of values that satisfy the condition given.

In this exercise, we see an inequality where we need to evaluate whether certain values for the variable make the inequality true or false. Understanding what each symbol represents allows us to determine whether any given solution fits the inequality. For instance, if a computation yields a number greater than zero, in the case of our original inequality, then it meets the inequality condition.

Having clarity on how inequalities function and apply helps students to effectively tackle a vast range of math problems.

Applying the Substitution Method Effectively

The substitution method is a handy technique in algebra to determine the truth value of an expression. By substituting specified values into the variables of an expression, you can discern the outcome of the inequality.

In the step-by-step solution, each value of x is tested by substituting it into the inequality. Here's how you can do it:

• Insert the given value into the variable in the inequality.
• Perform arithmetic operations (such as multiplication and addition) following the order of operations.
• Determine if the resultant expression satisfies the inequality condition.

This method is particularly useful as it provides a straightforward mechanism to check multiple potential solutions against the same inequality formula.

Enhancing Skills Through Mathematical Reasoning

Mathematical reasoning is the ability to think logically about the mathematics problem at hand and apply mathematical concepts to solve it. It’s what allows us to make informed decisions throughout the problem-solving process, such as in verifying if a substituted value makes an inequality true or false.

When tackling inequalities, we engage reasoning skills by interpreting each result accurately after substitution. We ensure our arithmetic operations were performed correctly and reflect whether the results align with expected outcomes.

Reasoning does not only involve computation but understanding the implications of each step. It also includes recognizing patterns and understanding the conditions imposed by inequalities. Mastering this aspect contributes significantly to overall mathematical proficiency.