Inequalities Worksheet featuring ten problems to solve, including linear and quadratic inequalities, from Math Monks.

Here are the step-by-step solutions for each inequality on the worksheet.

1) $6 - b < 8 - 4b$
* Add $4b$ to both sides to group the variables:
$6 + 3b < 8$
* Subtract $6$ from both sides:
$3b < 2$
* Divide by $3$:
$b < \frac{2}{3}$

2) $4(x + \frac{1}{2}) - 2(x + \frac{3}{2}) \le 5$
* Distribute the numbers into the parentheses:
$4x + 2 - 2x - 3 \le 5$
* Combine like terms ($4x - 2x = 2x$ and $2 - 3 = -1$):
$2x - 1 \le 5$
* Add $1$ to both sides:
$2x \le 6$
* Divide by $2$:
$x \le 3$

3) $-1 < x + 2 < 5$
* This is a compound inequality. We need to isolate $x$ in the middle.
* Subtract $2$ from all three parts:
$-1 - 2 < x < 5 - 2$
* Simplify:
$-3 < x < 3$

4) $3(y + 5) \le 2(y + 1)$
* Distribute the numbers:
$3y + 15 \le 2y + 2$
* Subtract $2y$ from both sides:
$y + 15 \le 2$
* Subtract $15$ from both sides:
$y \le -13$

5) $-5(u - 19) \le -6 + 2u$
* Distribute the $-5$:
$-5u + 95 \le -6 + 2u$
* Add $5u$ to both sides:
$95 \le -6 + 7u$
* Add $6$ to both sides:
$101 \le 7u$
* Divide by $7$:
$\frac{101}{7} \le u$ (or $u \ge \frac{101}{7}$)

6) $3 - 2(n - 4) > -1$
* Distribute the $-2$:
$3 - 2n + 8 > -1$
* Combine constants ($3 + 8 = 11$):
$11 - 2n > -1$
* Subtract $11$ from both sides:
$-2n > -12$
* Divide by $-2$. Remember: When dividing by a negative number, flip the inequality sign.
$n < 6$

7) $4r + 3 > 2r + 11$
* Subtract $2r$ from both sides:
$2r + 3 > 11$
* Subtract $3$ from both sides:
$2r > 8$
* Divide by $2$:
$r > 4$

8) $(x - 5)(x - 2) \le 0$
* Find the "critical points" where the expression equals zero: $x = 5$ and $x = 2$.
* Test the regions between and outside these points. Since we want the result to be less than or equal to zero (negative), we look for the region *between* the two numbers.
* Check a number between 2 and 5, like 3: $(3-5)(3-2) = (-2)(1) = -2$. This works.
* The solution includes the endpoints because of the $\le$ sign.
$2 \le x \le 5$

9) $x^2 + 7x + 10 < 10$
* Subtract $10$ from both sides to set it to zero:
$x^2 + 7x < 0$
* Factor out an $x$:
$x(x + 7) < 0$
* The critical points are $x = 0$ and $x = -7$.
* We need the expression to be negative. This happens between the two roots.
$-7 < x < 0$

10) $\frac{x^2 + 6}{2} \ge 53$
* Multiply both sides by $2$:
$x^2 + 6 \ge 106$
* Subtract $6$ from both sides:
$x^2 \ge 100$
* Take the square root. Remember that this creates two cases: one positive and one negative.
$x \ge 10$ or $x \le -10$

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Final Answer:
1) $b < \frac{2}{3}$
2) $x \le 3$
3) $-3 < x < 3$
4) $y \le -13$
5) $u \ge \frac{101}{7}$
6) $n < 6$
7) $r > 4$
8) $2 \le x \le 5$
9) $-7 < x < 0$
10) $x \ge 10$ or $x \le -10$