Question

Solve the inequality for x. Assume that a,band care positive constants.

$a\le bx+c<2a$

Short answer

The solution set of x is given by $\frac{a-c}{b}\le x<\frac{2a-c}{b}$.

Step-by-step solution

Step 1. Apply the concept of Solving linear inequalities.

An inequality looks like an equation, except that in the place of the equal sign is one of the symbols, $<,>,\le ,\ge$.

Example of an inequality is $4x+7\le 19$.

An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality sign.

Step 2. Simplify the inequality.

The solution set consists of all values of x that satisfy both of the inequalities $a\le bx+c\text{and}bx+c<2a$. Simplify the given inequality$a\le bx+c<2a$as follows:

localid="1646041969396" $\begin{array}{l}a-c\le bx+c-c<2a-c\text{(subtract}c)\\ a-c\le bx<2a-c\end{array}$

Step 3. Find the solution.

Isolate the variable x from$a-c\le bx<2a-c$

Divide throughout the inequality by b. Therefore,

$\begin{array}{l}\frac{a-c}{b}\le \frac{bx}{b}<\frac{2a-c}{b}\\ \frac{a-c}{b}\le x<\frac{2a-c}{b}\end{array}$