Math / Geometry

Question $\left\{ \begin{array}{l} x - y > 3 \\ x \geq 6 \end{array} \right.$ Graph system of inequalities to show all possible solutions.

Studdy Solution

STEP 1

What is this asking? We need to draw a picture that shows all the possible $x$ and $y$ values that make both $x - y > 3$ and $x \geq 6$ true! Watch out! Remember to check if the line itself is included in the solution by checking if the inequality includes the "equals" part (like $\geq$ or $\leq$ ).
Also, don't mix up which side to shade!

STEP 2

1. Graph the first inequality
2. Graph the second inequality
3. Find the overlapping region

STEP 3

Let's rewrite our first inequality, $x - y > 3$ , to make it easier to graph.
We want to get $y$ by itself.
We can add $y$ to both sides to get $x > 3 + y$ .
Then, subtract 3 from both sides to get $x - 3 > y$ , or $y < x - 3$ .

STEP 4

Now, we graph the line $y = x - 3$ .
This line has a slope of $1$ and a y-intercept of $-3$ .
Since our inequality is $y < x - 3$ and not $y \leq x - 3$ , the line itself is not included in the solution.
We represent this by drawing a dashed line.

STEP 5

To figure out which side of the line to shade, we can pick a test point that's not on the line, like $(0, 0)$ .
Plugging that into our inequality $y < x - 3$ , we get $0 < 0 - 3$ , or $0 < -3$ .
This is false!
So, we shade the side of the line that doesn't include $(0, 0)$ .

STEP 6

Our second inequality is $x \geq 6$ .
We graph the line $x = 6$ .
This is a vertical line passing through $x = 6$ .
Since the inequality is $\geq$ , the line is included in the solution.
We draw a solid line to show this.

STEP 7

Since we want $x$ values greater than or equal to 6, we shade the region to the right of the line $x = 6$ .

STEP 8

The solution to the system of inequalities is the region where the shading from both inequalities overlaps.
This is the region that satisfies both inequalities at the same time!

STEP 9

The solution is the region shaded by both $y < x - 3$ (below the dashed line) and $x \geq 6$ (to the right of the solid line).
It looks like a wedge pointing up and to the right, starting at the point $(6,3)$ but not including that point itself, since the line $y = x - 3$ is dashed.

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Question $\left\{ \begin{array}{l} x - y > 3 \\ x \geq 6 \end{array} \right.$ Graph system of inequalities to show all possible solutions.

Studdy Solution

STEP 1

STEP 2

1. Graph the first inequality
2. Graph the second inequality
3. Find the overlapping region

STEP 3

Let's rewrite our first inequality, $x - y > 3$ , to make it easier to graph.
We want to get $y$ by itself.
We can add $y$ to both sides to get $x > 3 + y$ .
Then, subtract 3 from both sides to get $x - 3 > y$ , or $y < x - 3$ .

STEP 4

Now, we graph the line $y = x - 3$ .
This line has a slope of $1$ and a y-intercept of $-3$ .
Since our inequality is $y < x - 3$ and not $y \leq x - 3$ , the line itself is not included in the solution.
We represent this by drawing a dashed line.

STEP 5

To figure out which side of the line to shade, we can pick a test point that's not on the line, like $(0, 0)$ .
Plugging that into our inequality $y < x - 3$ , we get $0 < 0 - 3$ , or $0 < -3$ .
This is false!
So, we shade the side of the line that doesn't include $(0, 0)$ .

STEP 6

Our second inequality is $x \geq 6$ .
We graph the line $x = 6$ .
This is a vertical line passing through $x = 6$ .
Since the inequality is $\geq$ , the line is included in the solution.
We draw a solid line to show this.

STEP 7

Since we want $x$ values greater than or equal to 6, we shade the region to the right of the line $x = 6$ .

STEP 8

The solution to the system of inequalities is the region where the shading from both inequalities overlaps.
This is the region that satisfies both inequalities at the same time!

STEP 9

The solution is the region shaded by both $y < x - 3$ (below the dashed line) and $x \geq 6$ (to the right of the solid line).
It looks like a wedge pointing up and to the right, starting at the point $(6,3)$ but not including that point itself, since the line $y = x - 3$ is dashed.

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Question{x−y>3x≥6\left\{ \begin{array}{l} x - y > 3 \\ x \geq 6 \end{array} \right.{x−y>3x≥6​ Graph system of inequalities to show all possible solutions.

Studdy Solution

STEP 1

STEP 2

1. Graph the first inequality2. Graph the second inequality3. Find the overlapping region

STEP 3

Let's **rewrite** our first inequality, x−y>3x - y > 3x−y>3, to make it easier to graph.We want to get yyy by itself.We can add yyy to both sides to get x>3+yx > 3 + yx>3+y.Then, subtract 3 from both sides to get x−3>yx - 3 > yx−3>y, or y<x−3y < x - 3y<x−3.

STEP 4

STEP 5

STEP 6

Our second inequality is x≥6x \geq 6x≥6.We **graph** the line x=6x = 6x=6.This is a **vertical line** passing through x=6x = 6x=6.Since the inequality is ≥\geq≥, the line *is* included in the solution.We draw a **solid line** to show this.

STEP 7

Since we want xxx values *greater than or equal to* 6, we shade the region to the *right* of the line x=6x = 6x=6.

STEP 8

The solution to the system of inequalities is the region where the shading from both inequalities *overlaps*.This is the region that satisfies *both* inequalities at the same time!

STEP 9

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Start learning now

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Question $\left\{ \begin{array}{l} x - y > 3 \\ x \geq 6 \end{array} \right.$ Graph system of inequalities to show all possible solutions.

1. Graph the first inequality
2. Graph the second inequality
3. Find the overlapping region

Let's rewrite our first inequality, $x - y > 3$ , to make it easier to graph.
We want to get $y$ by itself.
We can add $y$ to both sides to get $x > 3 + y$ .
Then, subtract 3 from both sides to get $x - 3 > y$ , or $y < x - 3$ .

Our second inequality is $x \geq 6$ .
We graph the line $x = 6$ .
This is a vertical line passing through $x = 6$ .
Since the inequality is $\geq$ , the line is included in the solution.
We draw a solid line to show this.

Since we want $x$ values greater than or equal to 6, we shade the region to the right of the line $x = 6$ .

The solution to the system of inequalities is the region where the shading from both inequalities overlaps.
This is the region that satisfies both inequalities at the same time!