Problem 14 - Elementary Algebra Within Reach

Question

Determine whether each value of \(x\) is a solution of the inequality. \(\frac{2}{3} x+4<6\) (a) \(x=7\) (b) \(x=0\) (c) \(x=-\frac{1}{2}\) (d) \(x=3\)

Step-by-Step Solution

Answer

The only values from given ones that satisfy the inequality \(\frac{2}{3} x+4<6\) are \(x=0\) and \(x=-1/2\).

Step 1: Substitute \(x=7\)

Begin by substituting \(x=7\) into the inequality \(\frac{2}{3} x+4<6\). This gives \(\frac{2}{3} * 7+4<6\). After performing the multiplication and addition on the left-hand side, we get \(14/3+4<6\), equivalently \(4.67+4<6\), and simplifying leads to \(8.67<6\). So, \(8.67<6\) is not a true statement. Therefore, \(x=7\) is not a solution to the inequality.

Step 2: Substitute \(x=0\)

Next, substitute \(x=0\). This gives \(\frac{2}{3} * 0+4<6\), which simplifies to \(0+4<6\) and thus \(4<6\). Here \(4<6\) is a true statement, proving that \(x=0\) is a solution to the inequality.

Step 3: Substitute \(x=-1/2\)

Now, substitute \(x=-1/2\). This gives \(\frac{2}{3} * -1/2+4<6\), which simplifies to \(-1/3+4<6\), equivalently \(1.33+4<6\), and simplifying leads to \(5.33<6\). Since \(5.33<6\) is true, it follows that \(x=-1/2\) is a solution of the inequality.

Step 4: Substitute \(x=3\)

Lastly, substitute \(x=3\). This gives \(\frac{2}{3} * 3+4<6\), which simplifies to \(2+4<6\) and thus \(6<6\). Here \(6<6\) is not a true statement, signifying that \(x=3\) is not a solution to the inequality.

Key Concepts

Algebraic Expressions

In algebra, expressions are formed using numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. An algebraic expression like \( \frac{2}{3} x + 4 \) includes a constant term (4) and a variable term (\( \frac{2}{3} x \)). The variable \( x \) can assume different values, and changing \( x \) alters the expression's outcome.

In the exercise, we examine each potential value of \( x \) to verify if it satisfies the inequality \( \frac{2}{3}{x} + 4 < 6 \). This inequality indicates that the collective result of the expression must be less than 6. A critical aspect of handling algebraic expressions is properly managing fractional coefficients like \( \frac{2}{3} \), ensuring accurate multiplication and addition with other terms.

Understanding how these components interact and resolve in inequalities helps in evaluating various values of the variable efficiently and identifying which ones meet the conditions set by the inequality.

Substitution Method

The substitution method involves replacing a variable within an expression or inequality with a specific number to determine the expression's value with that particular substitution. This process is straightforward and often used to validate possible solutions within algebraic equations or inequalities.

For instance, substituting different values for \( x \) such as \( x=0 \) or \( x=-\frac{1}{2} \) in the expression \( \frac{2}{3}x + 4 < 6 \) allows us to directly compute each case's validity. Once substituted, calculations are carried out:

• Multiply the fractional coefficient by the given value of \( x \).
• Add the result to the constant term in the expression.
• Compare this value to see if it meets the inequality condition (less than 6 in this case).

By carefully implementing substitution, we gain clarity regarding which values satisfy the inequality and which do not. This systematic approach simplifies the process and is especially helpful in checking multiple potential values swiftly.

Solution Verification

Verifying solutions is a vital process in solving inequalities as it confirms whether the investigated values truly satisfy the mathematical condition in question. After substitution, each computed result should be assessed for its truthfulness against the inequality condition.

For example, when evaluating \( x=-\frac{1}{2} \) in the inequality \( \frac{2}{3}x + 4 < 6 \), after performing the operations, we receive \( 5.33 < 6 \). This inequality holds true, thus \( x=-\frac{1}{2} \) is verified as a valid solution.

In contrast, substituting \( x=7 \) leads to the statement \( 8.67 < 6 \), which is incorrect, indicating that \( x=7 \) does not qualify as a solution.
By carrying out solution verification, we ensure the mathematical accuracy of our findings, thereby confirming the logical coherence of our calculated possibilities. This methodical approach ensures that no errors in inequality solution sets go unnoticed, reinforcing confidence in the solutions we identify.