Graph the system of inqualities presented here on your own paper can use your graph to answer the following questions: y>2x+3. Y<-3/2x-4

PART A.) Describe the graph of the system including shading in the types of lines graphed. provide a description of the solution area.

PART B.) is the point (-4,6) included in the solution area for the system? Justify your answer mathematically.

  A) see below for the graph

  B) no; it does not satisfy the second inequality

Step-by-step explanation:

It is useful to start by graphing the boundary line. Replace the inequality symbol with an equal sign, and you have the equation of that line.

When the equations are in slope-intercept form, as here, it is useful to start by plotting the y-intercept, then identifying additional points based on the slope.

y > 2x +3

The boundary line is dashed, because the > symbol does not include the "or equal to" case. The boundary line is y = 2x +3, a line with a y-intercept of 3 and a slope of 2. The slope of 2 means the line rises 2 units for each unit it goes to the right. Similarly, it falls 2 units for each unit to the left. Using this idea, you can plot the additional points (-2, -1) and (2, 7) on this (dashed) line.

The comparison symbol is >, and the location of y is to its left, meaning y > ... is the area you want to shade--the area above (greater than) the line.

__

y < -3/2x -4

As above, the boundary line is dashed, because the symbol is not ≤ or ≥. The y-intercept is -4, and the slope is -3/2. This slope means the line falls 3 units for each 2 units it goes to the right. Similarly, it rises 3 units for each 2 units to the left. Therefore a couple of additional points are (-2,-1) and (2, -7).

You may notice that the two lines have point (-2, -1) in common. This is where they intersect.

The comparison symbol is <, and y is to its left (y < ...) so the area you want to shade is y-values below (less than) the (dashed) line.

__

A) The graph is two dashed diagonal lines that intersect at (-2, -1). The solution area is to the left of their point of intersection, the doubly-shaded area between the diagonal lines. It does not include the lines themselves.

__

B) To show mathematically whether the point (-4, 6) is in the solution area, we can see if it satisfies both inequalities.

  y > 2x +3   ⇒   6 > 2(-4) +3   ⇒   6 > -5   TRUE

  y < -3/2x -4   ⇒   6 < -3/2(-4) -4   ⇒6 < 2   FALSE

The point is NOT included in the solution area. (The graph shows this as well.)

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: