Problem 56 - Elementary and Intermediate Algebra

Question

For each of the following, write a second inequality with the same meaning. $$ 12 \geq t $$

Step-by-Step Solution

Answer

t \, \leq 12

Step 1 - Understand the inequality

The inequality given is \[\begin{equation}12 \geq t\end{equation}\]This means that 12 is greater than or equal to t.

Step 2 - Think about the direction of the inequality

Inequalities can be rewritten by changing the sides of the equation and inverting the inequality sign. For example, \[\begin{equation}a \, \geq b\end{equation}\] is the same as \[\begin{equation}b \, \leq a.\end{equation}\]

Step 3 - Rewrite the inequality

Rewrite \[\begin{equation}12 \, \geq t \end{equation}\] as \[\begin{equation} t \, \leq 12\end{equation}\]

Key Concepts

Inequality Symbols

Inequality symbols are crucial in expressing the relationship between two values. The four most common inequality symbols are:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

These symbols help compare numbers and algebraic expressions.
For example, in the inequality \( 12 \geq t \) , the ≥ symbol indicates that 12 is greater than or equal to t. This relationship can be reversed by changing the sides and inverting the symbol, giving \( t \leq 12 \).
Understanding and properly using these symbols is essential for solving and rewriting inequalities effectively. Always ensure that the resulting inequality expresses the same relationship as the original.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to conclude mathematical problems. When dealing with inequalities, it's vital to maintain the relationship between the quantities involved.
Take the inequality \( 12 \geq t \) as an example. This statement tells us that 12 is either greater than or equal to t. To rewrite it, we need to preserve its meaning. We switch the sides and change the inequality sign to get \( t \leq 12 \).
It's essential to confirm that the new inequality is equivalent to the original one. Doing this requires a clear understanding of the order and properties of the inequality symbols. Practice and familiarity with these concepts improve your ability to apply mathematical reasoning when working with inequalities.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. When inequalities involve algebraic expressions, rewriting them may seem challenging, but the principles remain the same.
Consider the inequality \( 12 \geq t \), where t is a variable representing unknown values. This inequality signifies that 12 is greater than or equal to any value of t.
To rewrite it, swap the left-hand side and right-hand side while flipping the inequality sign: \( t \leq 12 \). The new inequality preserves the meaning of the expression.
Whether dealing with simple numbers or more complex algebraic expressions, the goal is always to present the same relationship in a clear and accurate form. Applying this practice helps students gain confidence in manipulating and understanding algebraic inequalities.