Answer:
[tex]\left[0,\dfrac{13}{2}\right)[/tex]
Step-by-step explanation:
Given compound inequality:
[tex]-7 < -2x+6\leq6[/tex]
To solve the compound inequality, we need to break it down into its individual inequalities, solve each one separately, and then combine their solutions.
Inequality 1:
-[tex]-7 < -2x+6\\\\\\-7-6 < -2x+6-6\\\\\\-13 < -2x\\\\\\\dfrac{-13}{-2} > \dfrac{-2x}{-2}\\\\\\\dfrac{13}{2} > x\\\\\\x < \dfrac{13}{2}[/tex]
Inequality 2:
[tex]-2x+6\leq 6\\\\\\-2x+6-6\leq 6-6\\\\\\-2x\leq0\\\\\\\dfrac{-2x}{-2}\geq \dfrac{0}{-2}\\\\\\x\geq 0[/tex]
Combine the intervals:
[tex]0\leq x < \dfrac{13}{2}[/tex]
In interval notation, this solution set is:
[tex]\Large\boxed{\boxed{\left[0,\dfrac{13}{2}\right)}}[/tex]
To graph this solution on a number line:
- Plot a closed circle at 0 (because 0 is included in the solution).
- Plot an open circle at 13/2 (because 13/2 is not included in the solution).
- Shade the region between 0 and 13/2 to represent all values of x that satisfy the inequality.
