Question

1. three times a number increased by four is less than -62.
2. the quotient of a number and 3 minus two is at least -12.

Explanation:

Problem 6

Step1: Define the variable

Let the number be $x$.

Step2: Translate the phrase

"Three times a number" is $3x$, "increased by four" is $+ 4$, "is less than -62" gives the inequality $3x + 4 < -62$.

Step3: Solve the inequality

Subtract 4 from both sides: $3x + 4 - 4 < -62 - 4$ → $3x < -66$.
Divide both sides by 3: $\frac{3x}{3} < \frac{-66}{3}$ → $x < -22$.

Problem 8

Step1: Define the variable

Let the number be $x$.

Step2: Translate the phrase

"The quotient of a number and 3" is $\frac{x}{3}$, "minus two" is $- 2$, "is at least -12" gives the inequality $\frac{x}{3}-2\geq - 12$.

Step3: Solve the inequality

Add 2 to both sides: $\frac{x}{3}-2 + 2\geq - 12 + 2$ → $\frac{x}{3}\geq - 10$.
Multiply both sides by 3: $\frac{x}{3}\times3\geq - 10\times3$ → $x\geq - 30$.

Answer:

$3x + 4 < -62$, $x < -22$