Question

directions: translate each expression.

1. “the sum of a number and 4.”
2. “the quotient of a number and -2.”
3. “the product of a number and 5.”
4. “the difference of a number and 7.”
5. “a number squared increased by 1”
6. “three less than twice a number”

directions: translate each equation. do not solve!

1. “twice a number divided by 6 is 42.”
2. “two times the sum of number and 5 is 20”
3. “four times the difference of a number and 7 is 32”
4. “one - third of a number is 12 less than the number itself”
5. “ten increased by the quotient of a number and 2 is -15”
6. “one less than the product of a number and 3 is 5 more than the number itself”

directions: translate each inequality.

1. “a number is less than 15”
2. “a number greater than or equal to -3”
3. “a number that is at least 90”
4. “a speed limit of 65 mph”
5. “a number that is no more than 20”
6. “a maximum number of 50”
7. “a minimum number of 10”
8. “a number no less than 25”

Explanation:

Step1: Define the number

Let the number be \(x\).

Step2: Translate expression 1

Sum means addition. So "the sum of a number and 4" is \(x + 4\).

Step3: Translate expression 2

Quotient means division. So "the quotient of a number and - 2" is \(\frac{x}{-2}\).

Step4: Translate expression 3

Product means multiplication. So "the product of a number and 5" is \(5x\).

Step5: Translate expression 4

Difference means subtraction. So "the difference of a number and 7" is \(x - 7\).

Step6: Translate expression 5

A number squared is \(x^{2}\), increased by 1 means addition. So it is \(x^{2}+1\).

Step7: Translate expression 6

Twice a number is \(2x\), three less than means subtraction. So it is \(2x - 3\).

Step8: Translate equation 7

Twice a number is \(2x\), divided by 6 gives \(\frac{2x}{6}\), and "is 42" means equal to 42. So the equation is \(\frac{2x}{6}=42\).

Step9: Translate equation 8

The sum of a number and 5 is \(x + 5\), two - times it is \(2(x + 5)\), and "is 20" means equal to 20. So the equation is \(2(x + 5)=20\).

Step10: Translate equation 9

The difference of a number and 7 is \(x - 7\), four times it is \(4(x - 7)\), and "is 32" means equal to 32. So the equation is \(4(x - 7)=32\).

Step11: Translate equation 10

One - third of a number is \(\frac{1}{3}x\), 12 less than the number itself is \(x-12\). So the equation is \(\frac{1}{3}x=x - 12\).

Step12: Translate equation 11

The quotient of a number and 2 is \(\frac{x}{2}\), ten increased by it is \(10+\frac{x}{2}\), and "is - 15" means equal to - 15. So the equation is \(10+\frac{x}{2}=-15\).

Step13: Translate equation 12

The product of a number and 3 is \(3x\), one less than it is \(3x - 1\), 5 more than the number itself is \(x + 5\). So the equation is \(3x-1=x + 5\).

Step14: Translate inequality 13

"A number is less than 15" is \(x<15\).

Step15: Translate inequality 14

"A number greater than or equal to - 3" is \(x\geq - 3\).

Step16: Translate inequality 15

"A number that is at least 90" means \(x\geq90\).

Step17: Translate inequality 16

A speed limit of 65 mph means the speed \(s\) satisfies \(s\leq65\). Let \(x\) represent speed, so \(x\leq65\).

Step18: Translate inequality 17

"A number that is no more than 20" is \(x\leq20\).

Step19: Translate inequality 18

"A maximum number of 50" is \(x\leq50\).

Step20: Translate inequality 19

"A minimum number of 10" is \(x\geq10\).

Step21: Translate inequality 20

"A number no less than 25" is \(x\geq25\).

Answer:

1. \(x + 4\)
2. \(\frac{x}{-2}\)
3. \(5x\)
4. \(x - 7\)
5. \(x^{2}+1\)
6. \(2x - 3\)
7. \(\frac{2x}{6}=42\)
8. \(2(x + 5)=20\)
9. \(4(x - 7)=32\)
10. \(\frac{1}{3}x=x - 12\)
11. \(10+\frac{x}{2}=-15\)
12. \(3x-1=x + 5\)
13. \(x<15\)
14. \(x\geq - 3\)
15. \(x\geq90\)
16. \(x\leq65\)
17. \(x\leq20\)
18. \(x\leq50\)
19. \(x\geq10\)
20. \(x\geq25\)