Solve the following inequality:
[tex]\lvert2x-1\rvert-3\leq4[/tex]As a first step we are going to add 3 to both sides:
[tex]\begin{gathered} \lvert2x-1\rvert-3+3\leq4+3 \\ \lvert2x-1\rvert\leq7 \end{gathered}[/tex]Then, we need to solve absolute value:
[tex]\begin{gathered} \lvert2x-1\rvert\leq7 \\ We\text{ know 2x-1}\leq7\text{ and 2x-1}\ge-7 \end{gathered}[/tex]Now, for the first inequality:
[tex]\begin{gathered} 2x-1\leq7 \\ 2x-1+1\leq7+1 \\ 2x\leq8 \\ x\leq\frac{8}{2} \\ x\leq4 \end{gathered}[/tex]For the second inequality:
[tex]\begin{gathered} 2x-1\ge-7 \\ 2x-1+1\ge-7+1 \\ 2x\ge-6 \\ x\ge-\frac{6}{2} \\ x\ge-3 \end{gathered}[/tex]The solution set would be, x≤4 and x≥−3, to represent on the number line:
