Question

Find the values of x that make the equation √(|x|+1) true.

Answer 1

The values of x that make the equation
`\( √(|x|+1) \)` true are
`\( x \geq -1 \)`and
`\( x \leq -1 \).`

Step-by-step explanation:

To find the values of x that satisfy the equation
`\( √(|x|+1) \)`, we need to consider the expression under the square root. The expression \( |x|+1 \) must be non-negative for the square root to be defined. Thus, we have two cases to consider:

1. When
`\( |x|+1 \geq 0 \):` This is always true for any value of x since the absolute value of any real number is non-negative. Therefore, there are no restrictions on x in this case.

2. When
`\( |x|+1 = 0 \):` This occurs only when
`\( x = -1 \)`, making the expression
`\( |x|+1 \)` equal to zero.

Combining these cases, the values of x that make the equation
`\( √(|x|+1) \)` true are
`\( x \geq -1 \) and \( x \leq -1 \).`

Understanding the conditions for the square root to be defined and the implications of the absolute value ensures a comprehensive analysis of the given equation. It is crucial to consider both cases to capture the complete range of x values that satisfy the equation.