Questions: For what values of the variable does each of the following expressions make sense? sqrt((1+3a)/25)

Solution Steps

To determine the values of the variable \( a \) for which the expression \(\sqrt{\frac{1+3a}{25}}\) makes sense, we need to ensure that the expression inside the square root is non-negative. This is because the square root function is only defined for non-negative numbers in the real number system.

1. Set the expression inside the square root to be greater than or equal to zero: \(\frac{1+3a}{25} \geq 0\).
2. Solve the inequality for \( a \).

To find the values of \( a \) for which the expression \( \sqrt{\frac{1+3a}{25}} \) is defined, we need to ensure that the expression inside the square root is non-negative: \[ \frac{1 + 3a}{25} \geq 0 \]

Multiplying both sides of the inequality by 25 (which is positive and does not change the direction of the inequality), we have: \[ 1 + 3a \geq 0 \]

Rearranging the inequality gives: \[ 3a \geq -1 \] Dividing both sides by 3 results in: \[ a \geq -\frac{1}{3} \]

The expression is defined for all values of \( a \) that satisfy the inequality \( a \geq -\frac{1}{3} \).

Final Answer