We need to solve the inequality:
[tex]4\left(2-x\right)>\left(5x-7\right)-\left(x-10\right)[/tex]
In order to do so, we can expand the expressions on each side, then apply the same operations on both sides of the inequality until we isolate the variable x and find the solution.
By expanding the expression, we obtain:
[tex]\begin{gathered} 4(2)+4(-x)>5x-7-x-(-10) \\ \\ 8-4x>5x-x-7+10 \\ \\ 8-4x\gt4x+3 \end{gathered}[/tex]
Now, adding 4x to both sides, we obtain:
[tex]\begin{gathered} 8-4x+4x>4x+3+4x \\ \\ 8>8x+3 \end{gathered}[/tex]
Subtracting 3 from both sides, we obtain:
[tex]\begin{gathered} 8-3>8x+3-3 \\ \\ 5>8x \end{gathered}[/tex]
Then, dividing both sides by 8, we obtain:
[tex]\begin{gathered} \frac{5}{8}>\frac{8x}{8} \\ \\ \frac{5}{8}>x \\ \\ x<\frac{5}{8} \end{gathered}[/tex]
Notice that x must be less than 5/8. Thus, 5/8 does not belong to the solution set. We represent this using (..., ...) for interval notation (open interval). Also, since there is no beginning to the interval solution, we write -∞ in replacement of the left boundary of the set.
Therefore, the solution set is:
Answer
[tex]\left(-\infty,\frac{5}{8}\right)[/tex]
