Chapter 1 - Solving Linear Equations - 1.4 - Solving Absolute Value Equations - Exercises - Page 34: 57

See explanation

For an absolute value equation $| x - a | = p$, where $a$ and $p$ are real numbers, the number of solutions for $x$ in this equation is dependent on the value $p$. a) If $p > 0$, then the equation has $2$ solutions. For example, let $a = 1$ and $p = 2$. Then the equation will be $| x - 1 | = 2$. i) Equating $x - 1 = 2$ adding $1$ to both sides, we get $x = 3$. ii) Equating $x - 1 = -2$ adding $1$ to both sides, we have $x = -1$. Thus $x$ can be $3$ or $-1$ . Therefore $2$ solutions. b) If $p = 0$, then equation has only one solution. For example, let $a = 1$. Since $p =0$, then the equation will be $| x - 1 | = 0$. i) Equating $x - 1 = 0$ adding 1 to both sides, we have $x = 1$. ii) Equating $x - 1 = -0 = 0$ (as $0$ does not take any sign) adding $1$ to both sides, we get $x = 1$. Thus $x$ will be $1$. Therefore only one solution. c) If $p<0$, then equation has no solution. This is because the absolute value can never. be negative. For example, let $a = 1$ and $p =-2$. Then the equation will be $| x - 1 | = -2$. Since an absolute value is always positive, there is no value for $x$ which can act as a solution for the equation. Thus, no solution.