Question 1

What is the least common denominator (LCD) of the fractions $$\frac{7x+1}{3}$$ and $$\frac{4x-5}{4}$$?

Accepted Answer

$$12$$

Question 2

After multiplying both sides of the inequality $$\frac{7x+1}{3} - \frac{4x-5}{4} \leq -2 by $$the LCD, which equation do you obtain?

Accepted Answer

$$4(7x+1) - 3(4x-5) \leq -24$$

Question 3

Which property allows you to subtract $$19$$ from both sides of the inequality $$16x + 19 \leq -24$$?

Accepted Answer

Addition property of inequality

Question 4

What is the solution to $$16x \leq -43$$?

Accepted Answer

$$x \leq -\frac{43}{16}$$

Question 5

How is the solution set $$x \leq -\frac{43}{16}$$ written in interval notation?

Accepted Answer

$$(-\infty, -\frac{43}{16}]$$

Question 6

Which step correctly applies the distributive property to $$4(7x+1) - 3(4x-5)$$?

Accepted Answer

$$28x + 4 - 12x + 15$$

Question 7

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

Accepted Answer

Reverse the inequality symbol

Question 8

Which of the following is a correct application of the addition property of inequality?

Accepted Answer

If $$a \u003c b$$, then $$a + c \u003c b + c$$

Question 9

A real-world scenario: A chemical process requires the temperature $$T (in $$degrees Celsius) to satisfy $$\frac{7T+1}{3} - \frac{4T-5}{4} \leq -2. $$What is the maximum allowable temperature for the process?

Accepted Answer

$$T = -\frac{43}{16}$$

Question 10

Which of the following intervals represents all possible solutions for $$x if$$ $$x$$ must be less than or equal to $$-\frac{43}{16}$$?

Accepted Answer

$$(-\infty, -\frac{43}{16}]$$

Question 11

Which property is used when combining $$28x + 4 - 12x + 15$$ into $$16x + 19$$?

Accepted Answer

Combining like terms

Question 12

Suppose a medical dosage $$d$$ must satisfy $$16d + 19 \leq -24. $$What is the maximum safe dosage?

Accepted Answer

$$d = -\frac{43}{16}$$

Question 13

If $$x$$ represents the number of hours a machine can run before maintenance, and $$x$$ must satisfy $$x \leq -\frac{43}{16}$$, what does this imply about the machine's operation?

Accepted Answer

The machine cannot run, since the number of hours cannot be negative.

Solving a Linear Inequality with Fractions and Writing the Solution in Interval Notation

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