Solve Linear Inequalities: example 2.75

Solve the inequality $8 x - 2 (5 - x) < 4 (x + 9) + 6 x$ , graph the solution on the number line, and write the solution in interval notation.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve the inequality $4 b - 3 (3 - b) > 5 (b - 6) + 2 b$ , graph the solution on the number line, and write the solution in interval notation.

How did it go?

Simplify each side as much as possible.	$8 x - 2 (5 - x) < 4 (x + 9) + 6 x$
Distribute.	$8 x - 10 + 2 x < 4 x + 36 + 6 x$
Combine like terms.	$10 x - 10 < 10 x + 36$
Subtract $10 x$ from both sides to collect the variables on the left.	$10 x - 10 - 10 x < 10 x + 36 - 10 x$
Simplify.	$−10 < 36$
The $x$ ’s are gone, and we have a true statement.	The inequality is an identity. The solution is all real numbers.
Graph the solution on the number line.
Write the solution in interval notation.	$(−∞, ∞)$

Simplify each side as much as possible.	$8 x - 2 (5 - x) < 4 (x + 9) + 6 x$
Distribute.	$8 x - 10 + 2 x < 4 x + 36 + 6 x$
Combine like terms.	$10 x - 10 < 10 x + 36$
Subtract $10 x$ from both sides to collect the variables on the left.	$10 x - 10 - 10 x < 10 x + 36 - 10 x$
Simplify.	$−10 < 36$
The $x$ ’s are gone, and we have a true statement.	The inequality is an identity. The solution is all real numbers.
Graph the solution on the number line.
Write the solution in interval notation.	$(−∞, ∞)$