Full solution

3x<5x+1<16

Break Compound Inequality: First, we will deal with the compound inequality by breaking it into two separate inequalities: $3x < 5x + 1$ and $5x + 1 < 16$ .
Solve $3x < 5x + 1$ : Now, let's solve the first inequality $3x < 5x + 1$ . We will subtract $5x$ from both sides to isolate the variable on one side. $\newline$ $3x - 5x < 5x + 1 - 5x$ $\newline$ $-2x < 1$
Solve $x > -\frac{1}{2}$ : Next, we divide both sides by $-2$ to solve for $x$ . Remember that dividing by a negative number reverses the inequality sign. $\newline$ $-2x / -2 > 1 / -2$ $\newline$ $x > -\frac{1}{2}$
Solve $5x + 1 < 16$ : Now, we will solve the second inequality $5x + 1 < 16$ . We will subtract $1$ from both sides to isolate the terms with $x$ . $\newline$ $5x + 1 - 1 < 16 - 1$ $\newline$ $5x < 15$
Solve $x < 3$ : We divide both sides by $5$ to solve for $x$ . $\frac{5x}{5} < \frac{15}{5}$ $x < 3$
Combine Inequalities: We now combine the results from the two inequalities to find the range of values for $x$ . The solution is the intersection of $x > -\frac{1}{2}$ and $x < 3$ . So, the final answer is $-\frac{1}{2} < x < 3$ .