I need help with solving inequalities?

Question

I need help on these following problems and how to graph them:

1) |x+1| ≥ 1.5

2) |-4 - (y-5)| - 3 > -1

For these questions, I need to answer and verify whether if it is true for all real numbers or has no solution:

1) -4 ≤ -1/2 - 2y ≤ 3/2

2) x+11 < -1 or -1/2x ≤ -4

3) -(8+m) < 8 - m

4) -3 -1/2t > 3 or 7.5 (1 - 8t) ≥ 63t

5) 2y + 3 < 4y - 1 ≤ 3 (y+1)

Answer 1

1) lx+1l ≥ 1.5  =&gt;  -1.5 ≥ x+1 or x+1 ≥ 1.5

So, it&#039;s -2.5 ≥ x or x ≥ 0.5

2) |-4 - (y-5)| - 3 &gt; -1  =&gt;  |-4 - (y-5)| &gt; 2  =&gt;  l1-yl &gt; 2

So, same as the first one. l1-yl &gt; 2  =&gt;  1 &gt; y or y &gt; 3

It&#039;s 1 &gt; y or y &gt; 3

Answer 2

When an inequality has an absolute value, you rewrite it as 2 inequalities.

1) |x+1| ≥ 1.5

x + 1 ≥ 1.5

x ≥ .5

-(x + 1) ≥ 1.5

-x - 1 ≥ 1.5

-x ≥ 2.5

x ≤ -2.5

To graph, draw circles above -2.5 and .5, fill in the circles. Draw a line from the circle above -2.5 going to the left. Draw a line from the circle above .5 going to the right.

2) |-4 - (y-5)| - 3 &gt; -1

(-4 - (y - 5)) - 3 &gt; -1

-4 - y + 5 - 3 &gt; -1

-y - 2 &gt; -1

-y &gt; 1

y &lt; -1

-(-4 - (y - 5)) - 3 &gt; -1

-(-4 - y + 5) - 3 &gt; -1

4 + y - 5 - 3 &gt; -1

y - 4 &gt; -1

y &gt; 3

Draw an open circle at y = -1, line going in the direction of -infinity.

Draw an open circle at y = 3, line going in the direction of infinity.

1) -4 ≤ -1/2 - 2y ≤ 3/2

Separate into two equations:

-4 ≤ -1/2 - 2y

Multiply by 2:

-8 ≤ -1 - 4y

-7 ≤ -4y

7/4 ≥ y

-1/2 - 2y ≤ 3/2

Multiply by 2:

-1 - 4y ≤ 3

-4y ≤ 4

y ≥ 1

Rewrite together (because they are overlapping solutions):

1 ≤ y ≤ 7/4

Draw filled in circles at 1 and 7/4, connect the circles with a line.

2) x + 11 &lt; -1 or -1/2x ≤ -4

Solve each separately:

x + 11 &lt; -1

x &lt; -12

-1/2x ≤ -4

x ≥ 8

Since the two solutions do not overlap, do not rewrite.

Draw an open circle at -12, line goes to the left.

Draw a filled in circle at 8, line goes to the right.

3) -(8 + m) &lt; 8 - m

-8 - m &lt; 8 - m

Add m to both sides:

-8 &lt; 8

True. The solution for this is ALL REAL NUMBERS.

4) -3 -1/2t &gt; 3 or 7.5 (1 - 8t) ≥ 63t

-3 - (1/2)t &gt; 3

-(1/2)t &gt; 6

t &lt; -12

7.5(1 - 8t) ≥ 63t

7.5 - 60t ≥ 63t

7.5 ≥ 123t

7.5/123 ≥ t

Draw a filled in circle at 7.5/123 (15/246), line to the left.

5) 2y + 3 &lt; 4y - 1 ≤ 3 (y+1)

Separate into two inequalities:

2y + 3 &lt; 4y - 1

2y + 4 &lt; 4y

4 &lt; 2y

2 &lt; y

4y - 1 ≤ 3(y + 1)

4y - 1 ≤ 3y + 3

y ≤ 4

Since the solutions overlap, rewrite:

2 &lt; y ≤ 4

Draw an open circle at 2 and a filled in circle at 4. Connect the circles with a line.