Can this system of inequalities be solved for x?

In summary, these two equations can be solved for x if you graph them and figure out where the regions of the two inequalities overlap.
  • #1
annamal
387
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Summary: Can these two equations be solved for x like a system of linear inequalities, and how?
##x- 2y \le 54##
##x + y \ge 93##

We start with
##x- 2y \le 54##
##x + y \ge 93##

Multiplying the second equation by 2, we have ##2x + 2y \ge 184##. We cannot seem to cancel the y out with the first equation because that would create an unclear inequality. So how do we solve it algebraically?

MENTOR NOTE: Moved to Precalculus Homework Help hence no template.
 
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  • #2
Draw the individual inequalities as straight lines in the xy-plane.
How can you use this to figure out when both your inequalities are satisfied?
Is there a single x-value, or does it depend on y?
 
  • #3
malawi_glenn said:
Draw the individual inequalities as straight lines in the xy-plane.
How can you use this to figure out when both your inequalities are satisfied?
Is there a single x-value, or does it depend on y?
That is solving it graphically. I would like to solve it algebraically.
 
  • #4
Start with the equalities first to find a common x,y that solves them.

BEFORE WE GO ANY FURTHER: PLEASE SHOW SOME WORK.
 
  • #5
The graphical solution will just be an aid for your algebra
 
  • #6
annamal said:
Summary: Can these two equations be solved for x like a system of linear inequalities, and how?
##x- 2y \le 54##
##x + y \ge 93##
You don't want to "solve" for x and y. Because these inequalities define two entire regions (half planes), not two thin lines with one intersection point.
y=13 and x=80 is the solution of x-2y=54; x+y=23. That doesn't tell you much.
Instead, you want to graph the lines x-2y=54; x+y=23 to see what the two regions (half planes) of the two inequalities are. Then you can see where the regions overlap. All the (x,y) points in the overlap are the answer.
 
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Related to Can this system of inequalities be solved for x?

1. Can all systems of inequalities be solved for x?

No, not all systems of inequalities can be solved for x. Some systems may have no solution, while others may have an infinite number of solutions.

2. How can I determine if a system of inequalities can be solved for x?

To determine if a system of inequalities can be solved for x, you can graph the inequalities and see if they intersect at a single point. If they do, then the system can be solved for x.

3. What is the process for solving a system of inequalities for x?

The process for solving a system of inequalities for x involves graphing the inequalities, identifying the point of intersection, and then solving for x using algebraic methods. This may include substitution or elimination.

4. Are there any special cases when solving a system of inequalities for x?

Yes, there are special cases when solving a system of inequalities for x. For example, if the inequalities are parallel, they have no solution. If they are the same line, they have an infinite number of solutions.

5. How can I check if my solution to a system of inequalities is correct?

You can check if your solution to a system of inequalities is correct by substituting the value of x into each inequality and verifying that it satisfies the inequality. If it does, then your solution is correct.

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