Chapter A, Problem 15E

To determine

To solve the below inequality in terms of intervals and illustrate the solution set on the real number line -

  $0\le 1-x<1$

Answer to Problem 15E

The solution of the inequality is $0<x\le 1$ and the solution set on a real number line -

  

Explanation of Solution

Given: Inequality: $0\le 1-x<1$

Formula Used:

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value.

Real number line is the line whose points are the real numbers. 

Calculation:

Given: Inequality equation is $0\le 1-x<1$

This inequality can be broken down into two inequalities:

  $0\le 1-x$ and $1-x<1$

Solving the firstinequality, we have:

Subtract $\left(-1\right)$ from both the sides -

  $0-1\le 1-x-1$

Solving further:

  $-1\le -x$

Divideboth the sides by $\left(-1\right)$ and reversing the inequality, we have:

  $\frac{-1}{-1}\ge \frac{-x}{-1}$

  $1\ge x$ ; i.e.:

  $x\le 1$

Solving the second inequality, we have:

  $1-x<1$

Subtract $\left(-1\right)$ from both the sides -

  $1-x-1<1-1$

Solving further:

  $-x<0$

Multiplyboth the sides by $\left(-1\right)$ and reversing the inequality, we have:

  $\left(-x\right)\times \left(-1\right)>0\times \left(-1\right)$

Solving further:

  $x>0$ ; i.e,

  $0<x$

Combining both the inequalities, we have:

  $0<x\le 1$

Drawing the above inequality on a real number line, we have:

  

Conclusion:

Hence, the solution of the inequality is $0<x\le 1$ and the solution set on a real number line -