Question 1

What is the least common denominator (LCD) of the fractions $$\frac{x+3}{16}$$, $$\frac{x-5}{10}$$, and $$\frac{1}{2}$$?

Accepted Answer

$$80$$

Question 2

After multiplying both sides of the inequality $$\frac{x+3}{16} + \frac{x-5}{10} \u003c \frac{1}{2} by $$the LCD, what is the resulting equation?

Accepted Answer

$$5(x+3) + 8(x-5) \u003c 40$$

Question 3

What is the solution set in interval notation for the inequality $$13x \u003c 65$$?

Accepted Answer

$$(-\infty, 5)$$

Question 4

Which step is necessary to clear the fractions in the inequality $$\frac{x+3}{16} + \frac{x-5}{10} \u003c \frac{1}{2}$$?

Accepted Answer

Multiply both sides by the LCD

Question 5

After distributing and combining like terms in $$5(x+3) + 8(x-5) \u003c 40$$, what is the simplified inequality?

Accepted Answer

$$13x - 25 \u003c 40$$

Question 6

If you solve $$13x - 25 \u003c 40$$, what is the next step?

Accepted Answer

Add $$25 to $$both sides

Question 7

What is the final solution for $$x in $$the inequality $$13x \u003c 65$$?

Accepted Answer

$$x \u003c 5$$

Question 8

Which property is used when $$5(x+3) + 8(x-5) is $$expanded to $$5x + 15 + 8x - 40$$?

Accepted Answer

Distributive property

Question 9

In the context of solving inequalities, what does interval notation $$(-\infty, 5)$$ represent?

Accepted Answer

All real numbers less than $$5$$

Question 10

If a solution set is written as $$\{x | x \u003c 5\}$$, what is the equivalent interval notation?

Accepted Answer

$$(-\infty, 5)$$

Question 11

Suppose you encounter the inequality $$\frac{x+3}{16} + \frac{x-5}{10} \u003c \frac{1}{2} in a $$real-world scenario, such as mixing chemicals where $$x$$ represents the amount of a substance. Why is it important to clear fractions before solving?

Accepted Answer

It simplifies calculations and makes the inequality easier to solve

Question 12

Which of the following is NOT a valid step when solving $$\frac{x+3}{16} + \frac{x-5}{10} \u003c \frac{1}{2}$$?

Accepted Answer

Adding the denominators together

Question 13

If you multiply both sides of an inequality by a negative number, what must you do to the inequality sign?

Accepted Answer

Reverse the direction of the inequality

Question 14

Which step comes immediately after combining like terms in the process of solving $$5x + 15 + 8x - 40 \u003c 40$$?

Accepted Answer

Add $$25 to $$both sides

Question 15

What is the correct solution set for the inequality $$x \u003c 5$$?

Accepted Answer

$$(-\infty, 5)$$

Solving Linear Inequalities with Fractions and Interval Notation

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