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