Question

Solve each equation. Check the solution.

61. If $\left|\text{x}+1\right|=\text{x}$

Short answer

There isno solution set for $\text{x}$.

Step-by-step solution

- State the definition of absolute value.

The distance of a number from 0 in the number line is called the absolute value and since distance cannot be negative so absolute value is always positive.

- Solve the inequality.

If x is a number, then absolute value of x is written as $\left|x\right|$.

Here, the inequality is:

$\left|x+1\right|=x$

Case 1:

$\begin{array}{c}a=b\\ x+1=x\\ x+1-1=x-1\\ x=x-1\end{array}$

Here value of cannot be determined.

Or Case 2:

$\begin{array}{c}a=-b\\ x+1=-x\\ x+1-1=-x-1\\ x+x=-x-1+x\\ 2x=-1\\ x=-0.5\end{array}$

The solution is $-0.5$. Here x has only one value.

- Check the solution.

For case 2: $x=-0.5$.

Check:

$\begin{array}{l}\left|x+1\right|=x\\ \left|-0.5+1\right|=-0.5\\ \left|0.5\right|=-0.5\\ \xcancel{0.5=-0.5}\end{array}$

Thus, it is not correct that is $x=-0.5$ is not a solution.

- State the conclusion.

Therefore, there is no solution set for x.