Can someone help with my math? 10 points?

Question

 

Write an inequality from the word problems and solve.
1. Jason has to write a report that is atleast 15 pages long. He has written 7 pages already. How many more pages will he need to write?
2. Tina has put 15 of her old CDs in a storage crate. How many more CDs can she store in the crate if it can hold no more than 23 Cds?
3. The product of a number and 3 increased by 2 is greater than or equal to 11.
4. Twice the sum of a number and -2 is less than or equal to 4.
5. The product of a number and 6 increased by 3 is greater than or equal to 15.
6. -8 is greater than 6 less than twice a number.

If anyone knows inequalities and can help with 6 short word problems please do so. Thanks alot

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Answer 1

1.   Let x be the number of pages Jasan need to write

     Jason has to write a report that is atleast 15 pages long and  he has written 7 pages already.

     So, x + 7 ≥ 15

    ⇒ x + 7 - 7 ≥ 15 - 7

    ⇒ x ≥ 8

    He need 8 or more than 8 pages to write.

Answer 2

2. Let x be the number cds that she can store in the crate.

   Tina has put 15 of her old CDs in a storage crate.

   The crate can hold no more than 23 cds.

   So, x + 15 ≤ 23

  ⇒ x + 15 -15 ≤ 23 - 15

  ⇒ x ≤ 8

  She can store 8 or less than 8 cds in the crate.

Answer 3

3) Let x  be a number .

Product of the number and 3 is 3x .

And incresed by 2 is greater than or equal to 11.

So it's algebraic expression is 3x + 2 ≥ 11

Subtract 2 from each side.

3x + 2 - 2 ≥ 11 - 2

3x ≥ 9

Divide each side by 3.

3x/3 ≥ 9/3

x ≥ 3

Solution {x|x ≥ 3}

4) Let x be the number.

Twice the sum of the number and -2 is less than or equal to 4.

It's algebraic expression is 2x - 2 ≤ 4

Add 2 to each side .

2x - 2 + 2 ≤ 4 + 2

2x ≤ 6

Divide each side by 2.

2x/2 ≤ 6/2

x ≤ 3

Solution {x|x ≤ 3}

5) Let x be a number.

The product of the number and 6 increased by 3 is greater than or equal to 15.

6x + 3 ≥ 15

Subtract 3 from each side.

6x + 3 - 3 ≥ 15 - 3

6x ≥ 12

Divide each side by 6.

6x/6 ≥ 12/6

x ≥ 2

Solution {x| x ≥ 2}

6) Let x be a number.

-8 is greater than 6 less than twice the number.

It's algebraic expreession is -8 > 2x < 6

To solve the compound inequality -8 > 2x < 6

Solve the inequality1) :  -8 > 2x

-8/2 > x

-4 > x

x < -4

Solve the inequality 2) :  2x < 6

x < 6/2

x < 3

solution x < 3 and x < -4

The above solution can be written as x < -4

solution { x| x < -4}.

Observe the graph the open circle means -4 is not included in the solution set.

Answer 4

4) Sum of a number and -2 is x  - 2.

Twice the sum of a number and -2 is 2(x  - 2).

Twice the sum of a number and -2 is less than or equal to 4 is 2(x  - 2) ≤ 4.

Apply Distributive Property: a(b - c) = ab - ac.

2(x) - 2(2) ≤ 4

2x - 4 ≤ 4

Add 4 to each side.

Apply Addition Property of inequality: If a ≤ b then a + c ≤ b + c.

2x - 4 + 4 ≤ 4 + 4

Apply Additive Inverse Property: -a + a = 0.

2x + 0 ≤ 4 + 4

Apply Additive Identity Property: a + 0 = a.

2x ≤ 4 + 4

2x ≤ 8

Divide each side by 2.

Apply Division Property of inequality: If a ≤ b then a / c ≤ b / c.

2x / 2 ≤ 8 / 2

Cancel common terms.

x  ≤ 8 / 2

x  ≤ 4

The solution is x  ≤ 4.

Answer 5

6) Twice the number is 2x.

6 less than twice the number is 2x - 6.

"-8 is greater than 6 less than twice a number" is -8 > 2x - 6.

Add 6 to each side.

Apply Addition Property of inequality: If a > b then a + c > b + c.

-8 + 6 > 2x - 6 + 6

-2 > 2x - 6 + 6

Apply Additive Inverse Property: -a + a = 0.

-2 > 2x + 0

Apply Additive Identity Property: a + 0 = a.

-2 > 2x

Divide each side by 2.

Apply Division Property of inequality: If a > b then a / c > b / c.

-2 / 2 > 2x / 2

Cancel common terms.

-1 > x.

=> x < -1.

Therefore the solution is x < -1.