Solving and graphing inequalities?

Question

Once again, I am stumped by the abundance of algebra 2 problems. How do I solve the open sentence and graph the solution set on a number line?:

1) |4d + 7| = 5

2) |2/3 k - 1| ≥ 2

3) 3 > |a + 1|

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These are optional, but if you'd like to solve them, go ahead.

4) |x + 2| + |x - 2| = 4

5) |x + 2| - |x - 2| = 4

Thanks in advance.

Answer 1

1) |4d + 7| = 5

(4d + 7) = 5

4d = -2

d = -1/2

-(4d + 7) = 5

4d + 7 = -5

4d = -12

d = -3

Draw a solid dot at -3 and -1/2

2) |2/3 k - 1| ≥ 2

(2/3)k - 1 ≥ 2

(2/3)k ≥ 2

k ≥ 2(3/2)

k ≥ 3

-((2/3)k -1) ≥ 2

-(2/3)k + 1 ≥ 2

-(2/3)k ≥ 1

-1 ≥ (2/3)k

(3/2)(-1) ≥ k

-3/2 ≥ k

Solutions do not overlap. Draw filled in circle at -3/2, line going to left. Draw a filled in circle at 3, line going to right.

3) 3 > |a + 1|

3 > a + 1

2 > a

3 > -(a + 1)

3 > -a - 1

a > -4

Overlapping solutions: -4 < a < 2

Draw open circles at -4 and 2, connect them with a line.

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These are optional, but if you'd like to solve them, go ahead.

4) |x + 2| + |x - 2| = 4  (2 absoulute value terms, 4 possible solutions)

x + 2 + x - 2 = 4

2x = 4

x = 2

-(x + 2) + x - 2 = 4

-x - 2 + x - 2 = 4

-4 = 4 FALSE

x + 2 - (x - 2) = 4

x + 2 - x - 2 = 4

0 = 4 FALSE

-(x + 2) - (x - 2) = 4

-x - 2 - x + 2 = 4

-2x = 4

x = -2

Only two real solutions: x = 2, -2. Draw dots at -2, 2.

5) |x + 2| - |x - 2| = 4

x + 2 - (x - 2) = 4

x + 2 - x + 2 = 4

4 = 4

Solution: ALL REAL NUMBERS

-(x + 2) - (x - 2) = 4

-x - 2 - x - 2 = 4

-2x - 4 = 4

-2x = 8

x = -4

x + 2 - -(x - 2) = 4

x + 2 + (x - 2) = 4

x + 2 + x - 2 = 4

2x = 4

x = 2

-(x + 2) - -(x - 2) = 4

-x - 2 + (x - 2) = 4

-x - 2 + x - 2 = 4

-4 = 4 FALSE (no solution)

Solutions: All real numbers, when applying the absolute value does not change the solution. -4 when applying absolute value to the first term only and 2 when applying it to the second term only.

Answer 2

!) 4d+7= +_5    

   4d=+_5-7

     Two answers: d1=-2    d2=-12  

2) 2/3k-1>=2      

  2/3k>=3

   k>=9/2

   2/3k-1<= - 2

   2/3k<=-1

    k<=-3/2

  Answer:   k<=-3/2 and k>=9/2

3)   -3  <a+1< +3

   - 4<a<2 this is the answer

4)  Ther are three steps and u know why

x>=2:  x+2+x-2=4    x=2 (is right)

-2<x<2     x+2-x+2+4     4=4   ( all the range -2<x<2 is right!!!)



x<=-2        -x-2-x+2=4     x=-2  (is alright)

Answer:   - 2<=x<= +2

5) Again three steps

x>=2        x+2-x+2=4    4=4   (the whole range x>=2 fits)

-2<x<+2   x+2+x-2=4     2x=4 x=2 (out of the rang)

x<=-2    -x-2+x-2=4    -4=4 (no x for it)

Answer:  x>=2

  

Answer 3

1). |4d + 7| = 5

|      4d+7|^2 = 5^2

16d^2+56d+49 = 25

16d^2+56d+49-25 = 0

16d^2+56d+24 = 0

2d^2+7d+3 = 0

  d = -7(+-)√(7^2-4*2*3)/(2*2)

  d = [-7(+-)5]/4

  d = -3 or d=-1/2.

2).

|2/3 k - 1| ≥ 2

|2/3k-1|^2 ≥ 2^2

4/9k^2-4/3k+1≥4

4/9k^2-4/3k-3≥0

k^2-3k-(27/4)≥0

At x-axis, when f(x)=0,

k = -(-3)(+-)√(3^2-4*1*(-27/4)/(2*1)

k = [3(+-)6]/2

k = -1.5 or k=4.5

Therefore, k<-1.5 or k>4.5.

3) 3>|a + 1|

|a+1|<3

|a+1|^2<3^2

a^2+2a+1<9

a^2+2a-8<0

(a+4)(a-2)<0

At x-axis, when f(x) =0,

   (a+4)(a-2) = 0

     a=2 or a=-4

Therefore, x>-4 or x<2.

Hope this helps.