Question

solve the inequality \\(-\frac{5}{3}\left(\frac{9}{10}x + 15\
ight) < 7 - \left(8 - \frac{9}{2}x\
ight)\\) and express the solution in interval notation. there is also a number line with marks at -8, -6, -4, -2, 0, 2, 4, 6, 8 for interval notation representation.

Explanation:

Step1: Simplify both sides

First, distribute \(-\frac{5}{3}\) on the left: \(-\frac{5}{3} \cdot \frac{9}{10}x - \frac{5}{3} \cdot 15 = -\frac{3}{2}x - 25\).
On the right, simplify \(7 - (8 - \frac{9}{2}x) = 7 - 8 + \frac{9}{2}x = -1 + \frac{9}{2}x\).
Now the inequality is \(-\frac{3}{2}x - 25 < -1 + \frac{9}{2}x\).

Step2: Move x terms to one side

Add \(\frac{3}{2}x\) to both sides: \(-25 < -1 + 6x\).

Step3: Isolate x term

Add 1 to both sides: \(-24 < 6x\).

Step4: Solve for x

Divide both sides by 6: \(-4 < x\), or \(x > -4\).

Answer:

The solution in interval notation is \((-4, \infty)\)