How do you solve and plot inequalities with multiple variables?

In summary, you need to solve the inequalities 2<x<6, 1<y<5, y-2≤2x, -2y≥8-4x and plot them on a graph to find the common area. To do this, you will graph the lines x=2 and x=6 with dashed lines, divide the plane into three regions, and shade the region between 2 and 6 since it satisfies both inequalities. This will give you the desired shaded region on the graph.
  • #1
ai93
54
0
I have the inequalities \(\displaystyle 2<x<6,\quad 1<y<5,\quad y-2\le2x, \quad-2y\ge8-4x\) I have to solve these and plot it in a graph and show the region where they satisfy. I understand you have to find the common area and shade it.

How do you find the points to plot for \(\displaystyle 2<x<6\quad and\quad 1<y<5\)

I think I have solved \(\displaystyle y-2\le2x\quad and -2y\ge8-4x\) to \(\displaystyle y\le2x+2\quad and \quad y\le-4+2x\) I am just unsure on the first two. Will making a X and Y table help to find the points?
 
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  • #2
Well, the region $2<x<6$ contains all the points in the $x,y$-plane where the $x$-component lies between $2$ and $6$ (both not included).
 
  • #3
Siron said:
Well, the region $2<x<6$ contains all the points in the $x,y$-plane where the $x$-component lies between $2$ and $6$ (both not included).

If i made a graph, would it go through the $y$ axis diagonal through the points $2$ and $6$?
 
  • #4
mathsheadache said:
If i made a graph, would it go through the $y$ axis diagonal through the points $2$ and $6$?

For the inequality $2<x<6$, I would begin by graphing the lines $x=2$ and $x=6$ with dashed lines since the inequality is strict on both sides of $x$. Now you have divided the plane into 3 regions. which of these regions should you shade?
 
  • #5
MarkFL said:
For the inequality $2<x<6$, I would begin by graphing the lines $x=2$ and $x=6$ with dashed lines since the inequality is strict on both sides of $x$. Now you have divided the plane into 3 regions. which of these regions should you shade?

Because it is x>2 is greater than, I shade everything to the right. And becuase x<6 is less than, shade everything to the left? I will be left with a shaded region in between 2 and 6?
 
  • #6
mathsheadache said:
Because it is x>2 is greater than, I shade everything to the right. And becuase x<6 is less than, shade everything to the left? I will be left with a shaded region in between 2 and 6?

Yes, good! (Sun)

That's what $2<x<6$ means...any value of $x$ on the interval $(2,6)$, i.e., any value of $x$ in between 2 and 6, but not including 2 and 6. :D

So, the region you have shaded contains all the points in the plane for which the $x$-coordinate satisfies the given compound inequality.
 
  • #7

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FAQ: How do you solve and plot inequalities with multiple variables?

What are inequalities?

Inequalities are mathematical statements that compare two quantities or expressions. They use symbols such as <, >, ≤, and ≥ to indicate the relationship between the two quantities.

How are inequalities represented on a graph?

Inequalities can be graphed on a number line or coordinate plane. The symbols used in the inequality determine whether the line or shaded region is open or closed. For example, if the symbol is <, the line will be dashed and the shaded region will be above the line. If the symbol is ≥, the line will be solid and the shaded region will be below the line.

What is the difference between an equation and an inequality?

An equation is a statement that shows that two expressions are equal. Inequalities, on the other hand, show the relationship between two expressions is not equal. Inequalities can also have more than one solution, while equations only have one solution.

How do you solve inequalities?

To solve an inequality, you must isolate the variable on one side of the inequality sign, similar to solving an equation. However, if you multiply or divide by a negative number, you must flip the inequality sign. The solution to an inequality is a range of values that make the inequality true.

What is the purpose of using inequalities in real-world situations?

Inequalities are used to represent constraints or limits in real-world situations. They can be used to represent things like budgets, age restrictions, and minimum or maximum values. Inequalities also help us make decisions by showing us what options are available or not available.

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