Solve the following inequality. Justify each step.

negative 2 less than or equals StartFraction t Over 4 EndFraction
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Part 1
What is the​ solution?
  
t greater than or equals minus 8
​(Simplify your answer. Type an​ inequality.)
Part 2
Select all properties of inequalities used to obtain the solution.
A.
Multiplication property of inequality
B.
Addition property of inequality
C.
Subtraction property of inequality
D.
Division property of inequality

1 answer

Let’s solve the inequality step by step:

Given Inequality:

\[ -2 \leq \frac{t}{4} \]

Part 1: Solve for \( t \)

  1. Multiply both sides by 4:
    • To eliminate the fraction, we can multiply both sides of the inequality by 4. Since 4 is a positive number, the direction of the inequality will remain the same. \[ -2 \cdot 4 \leq \frac{t}{4} \cdot 4 \] This simplifies to: \[ -8 \leq t \] or equivalently: \[ t \geq -8 \]

Solution:

\[ t \geq -8 \]

Part 2: Properties of Inequalities

In solving the inequality, we used:

A. Multiplication property of inequality (specifically, we multiplied both sides by 4)

The other properties (B, C, and D) were not used, as we did not add, subtract, or divide by any number.

Thus, the correct selection for the property used is:

  • A. Multiplication property of inequality