Question

Explain how to solve the given problem.

(a) Linear inequality: $2x\ge 1$

(b) Nonlinear inequality: $\left(x-1\right)\left(x-4\right)<0$

(c) Absolute value equation: $\left|2x-5\right|=7$

(d) Absolute value inequality:$\left|2x-5\right|\le 7$

Short answer

(a) The solution of linear inequality is$x=\left[\frac{1}{2},\infty \right)$.

(b) The solution of the non linear inequality is $x=\left(1,4\right)$.

(c)The solution of absolute value equation is $x=6,-1$.

(d)The solution of absolute value inequality is $x=\left[-1,6\right]$.

Step-by-step solution

Part a. Step 1. Write the given linear inequality.

The given linear inequality is,

$2x\ge 1$

Part a. Step 2. Make the coefficient of the variable term 1.

Divide both side of equation (1) by 2.

$$\begin{gathered} \frac{2}{2}x\ge \frac{1}{2} \\ x\ge \frac{1}{2} \end{gathered}$$

Part a. Step 3. Express the inequality in terms of interval.

The inequality in terms of interval may be expressed as,

$x=\left[\frac{1}{2},\infty \right)$

The inequality include$\frac{1}{2}$ so use square bracket and use a open bracket for $\infty$.

Thus, the solution of linear inequality is $x=\left[\frac{1}{2},\infty \right)$.

Part b. Step 1. Write the given nonlinear inequality.

The given nonlinear inequality is,

$\left(x-1\right)\left(x-4\right)<0$

Part b. Step 2. Set the factor equal to zero.

Apply Zero product property

$$\begin{gathered} \left(x-1\right)<0 \\ x-1<0 \\ x<1 \end{gathered}$$

And,

$$\begin{gathered} x-4<0 \\ x<4 \end{gathered}$$

Part b. Step 3. Express the inequality in terms of interval.

The inequality in terms of interval may be expressed as,

$x=\left(1,4\right)$

Thus, the solution set is $x=\left(1,4\right)$.

Part c. Step 1. Write the given expression.

The given expression is,

$\left|2x-5\right|=7$

Part c. Step 2. Define absolute function.

For a real number x, its absolute value is defined as,

$\left|x\right|=\left\{\begin{array}{l}x,\text{\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if}x\ge 0\\ -x,\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if}x<0\end{array}\right.$

This means,$\left|x\right|$ always gives values greater than or equal to zero.

Part c. Step 3. Solve the expression.

The above expression is solved as,

$$\begin{gathered} 2x-5=7 \\ x=\frac{12}{2} \\ =6 \end{gathered}$$

And,

$$\begin{gathered} 2x-5=-7 \\ x=\frac{-2}{2} \\ =-1 \end{gathered}$$

Thus, the solution of the equation is$x=6,-1$

Part d. Step 1. Write the given expression.

The given expression is,

$\left|2x-5\right|\le 7$

Part d. Step 2. Define absolute function.

For a real number x, its absolute value is defined as,

$\left|x\right|=\left\{\begin{array}{l}x,\text{\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if}x\ge 0\\ -x,\text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if}x<0\end{array}\right.$

This means,$\left|x\right|$ always gives values greater than or equal to zero.

Part d. Step 3. Solve the expression.

The above expression is solved as,

$$\begin{gathered} 2x-5=7 \\ x=\frac{12}{2} \\ =6 \end{gathered}$$

And,

$$\begin{gathered} 2x-5=-7 \\ x=\frac{-2}{2} \\ =-1 \end{gathered}$$

The values in the inequality form is,

$-1\le x\le 6$

Thus, the solution of the expression is$\left[-1,6\right]$