Answer :
We interpret the question as follows:
[tex]2|x-5|-6\ge14[/tex]To answer this inequality, we can proceed as follows:
1. Add 6 to both sides of the inequality:
[tex]2|x-5|-6+6\ge14+6\Rightarrow2|x-5|\ge20[/tex]2. Divide both sides of the inequality by 2:
[tex]\frac{2}{2}|x-5|\ge\frac{20}{2}\Rightarrow|x-5|\ge10[/tex]Then, using a property of inequalities when is present the absolute value function, we have:
[tex]|a|\ge b\Leftrightarrow a\leq-b\text{ or a}\ge b[/tex]Now, we can apply it as follows:
[tex]x-5\leq-10\text{ or x-5}\ge10[/tex]And we have to solve both inequalities separately as follows:
[tex]x-5\leq-10\Rightarrow x-5+5\leq-10+5\Rightarrow x\leq-5[/tex]And
[tex]x-5\ge10\Rightarrow x-5+5\ge10+5\Rightarrow x\ge15[/tex]We can graph both solutions as follows:
We can see that above the numbers, we have either a [ or ], or a solid point to represent that the number is included in the solution.
In summary, therefore, the solution for the inequality above is:
[tex]x\leq-5\text{ or x}\ge15[/tex]We can also represent this solution in interval form as follows:
[tex](-\infty,-5\rbrack\cup\lbrack15,\infty)[/tex]
