Solve the following equation, write down the solution set and represent the solution set on the real number line. $- 2 + 10 x \leq 13 x + 10 < 24 + 10 x, x \in Z$ .

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Solution

Step 1 (Solve the inequation):
$- 2 + 10 x \leq 13 x + 10 < 24 + 10 x, x \in Z$
subtracting $10 x$ to both sides
$- 2 \leq 3 x + 10 < 24 - 12 \leq 3 x < 14 x \geq - 4 a n d x < \frac{14}{3}$
Step2 (Representation on the number line):
The solution set will be $- 4 \leq x < \frac{14}{3}$ , and the solution set includes $x = {- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4}$ so the solution set on the number line will be
Hence the solution set of the inequality $- 2 + 10 x \leq 13 x + 10 < 24 + 10 x, x \in Z$ is $x = {- 4, - 3, - 2, - 1, 0, 1, 2, 3, 4}$ and the representation of the solution set on the number line is as shown above.

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