Which is true about the solution to the system of inequalities shown?y < One-thirdx – 1y < One-thirdx – 3On a coordinate plane, 2 solid straight lines are shown. The first line has a positive slope and goes through (0, negative 1) and (2, 0). Everything below the line is shaded. The second line has a positive slope and goes through (0, negative 3) and (3, negative 2). Everything below the line is shaded. All values that satisfy y < One-thirdx – 1 are solutions.All values that satisfy y < One-thirdx – 3 are solutions.All values that satisfy either y < One-thirdx – 1 or y < One-thirdx – 3 are solutions.There are no solutions.

Answer 1

To analyze the system of inequalities described, let's break it down step by step:

1. Understanding the Inequalities:

   • The first inequality is \( y < \frac{1}{3}x - 1 \), which represents a region below the line with a slope of \( \frac{1}{3} \) that intersects the y-axis at -1.
   • The second inequality is \( y < \frac{1}{3}x - 3 \), which represents a region below another line with the same slope of \( \frac{1}{3} \) that intersects the y-axis at -3.

2. Graphing the Lines:

   • The first line, \( y = \frac{1}{3}x - 1 \), passes through the points (0, -1) and (2, 0).
   • The second line, \( y = \frac{1}{3}x - 3 \), passes through the points (0, -3) and (3, -2).

3. Shading the Regions:

   • For both inequalities, the shading is below each line.

4. Interpreting the Solutions:

   • Since both inequalities are "less than," any point below either line will satisfy at least one of the inequalities.
   • The solution set includes all points that satisfy either \( y < \frac{1}{3}x - 1 \) or \( y < \frac{1}{3}x - 3 \).

5. Conclusion:

   • It is true that all values that satisfy either of the inequalities will be solutions to the system because the two lines do not contradict each other; they create overlapping regions.

Thus, the correct statement is: All values that satisfy either \( y < \frac{1}{3}x - 1 \) or \( y < \frac{1}{3}x - 3 \) are solutions.

Answer 2

Which system of linear inequalities is represented by the graph?y > x – 2 and y < x + 1
y < x – 2 and y > x + 1
y < x – 2 and y > x + 1
y > x – 2 and y < x + 1

Answer 3

To determine which system of linear inequalities is represented by the graph, we need to consider the behavior of each inequality in the context of their respective regions on a coordinate plane.

1. Understanding the Inequalities:

   • Inequalities Provided:
     • \( y > x - 2 \): This inequality represents the region above the line \( y = x - 2 \).
     • \( y < x + 1 \): This inequality represents the region below the line \( y = x + 1 \).
   • The other options similarly represent different regions based on their respective inequalities.

2. Analyzing Each Option:

   • Option 1: \( y > x - 2 \) and \( y < x + 1 \)
     • This would represent a region between the lines \( y = x - 2 \) (above this line) and \( y = x + 1 \) (below this line).
   • Option 2: \( y < x - 2 \) and \( y > x + 1 \)
     • This would indicate regions below \( y = x - 2 \) and above \( y = x + 1 \), which means the two regions do not overlap and thus would not represent a continuous area.
   • Option 3: \( y < x - 2 \) and \( y > x + 1 \)
     • This suggests being below \( y = x - 2 \) and above \( y = x + 1\), which also does not create a valid solution region.
   • Option 4: \( y > x - 2 \) and \( y < x + 1 \)
     • This indicates an area above \( y = x - 2 \) and below \( y = x + 1\).

3. Conclusion:

   • The only option that describes a valid region where there are specific inequalities that clearly intersect, creating a bounded area between the two lines, is:

\( y > x - 2 \) and \( y < x + 1 \).

Therefore, this is the correct representation of the graph of linear inequalities.

Answer 4

Which ordered pair makes both inequalities true? y < –x + 1y > xOn a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (0, 1) and (1, 0). Everything below and to the left of the line is shaded. The second dashed line has a positive slope and goes through (negative 1, negative 1) and (1, 1). Everything above and to the left of the line is shaded. (–3, 5)(–2, 2)(–1, –3)(0, –1)

Answer 5

To determine which ordered pair makes both inequalities true for the given inequalities:

1. Inequalities Given:

   • \( y < -x + 1 \)
   • \( y > x \)

2. Understanding Each Inequality:

   • For the first inequality \( y < -x + 1 \):
     • The line \( y = -x + 1 \) has a y-intercept of 1 and a slope of -1. The solution region is below this line.
   • For the second inequality \( y > x \):
     • The line \( y = x \) has a y-intercept of 0 and a slope of 1. The solution region is above this line.

3. Visualizing the Regions:

   • The region below the line \( y = -x + 1 \) includes points that are to the left of and below the line.
   • The region above the line \( y = x \) includes points that are above the line.

4. Evaluating the Ordered Pairs: Now, we will check each of the provided ordered pairs against both inequalities:

   • (–3, 5):

     • For \( y < -x + 1 \): \( 5 < -(-3) + 1 \rightarrow 5 < 4\), False.
     • For \( y > x \): \( 5 > -3\), True.
     • Not a solution.

   • (–2, 2):

     • For \( y < -x + 1 \): \( 2 < -(-2) + 1 \rightarrow 2 < 3\), True.
     • For \( y > x \): \( 2 > -2\), True.
     • This pair satisfies both inequalities.

   • (–1, –3):

     • For \( y < -x + 1 \): \( -3 < -(-1) + 1 \rightarrow -3 < 2\), True.
     • For \( y > x \): \( -3 > -1\), False.
     • Not a solution.

   • (0, –1):

     • For \( y < -x + 1 \): \( -1 < -0 + 1 \rightarrow -1 < 1\), True.
     • For \( y > x \): \( -1 > 0\), False.
     • Not a solution.

5. Conclusion: The ordered pair (–2, 2) satisfies both inequalities \( y < -x + 1 \) and \( y > x \). Thus, it is the ordered pair that makes both inequalities true.

Answer 6

Which system of linear inequalities has the point (3, –2) in its solution set?y less-than negative 3. y less-than-or-equal-to two-thirds x minus 4 1. On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded. y greater-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 2. On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded. y less-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 3. On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded. y greater-than negative 2. y less-than-or-equal-to two-thirds x minus 4 4. On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal at y = negative 2. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.

Answer 7

To determine which system of linear inequalities has the point (3, -2) in its solution set, we first need to evaluate the coordinates of the point against each inequality in the provided options.

Given Point:

The point is (3, -2), which means:

• \( x = 3 \)
• \( y = -2 \)

Inequalities to Evaluate:

1. Option 1:
   \( y < -3 \)
   \( y \leq \frac{2}{3}x - 4 \)

   • For \( y < -3 \):
     \(-2 < -3\) → False
   • For \( y \leq \frac{2}{3}(3) - 4\):
     \(-2 \leq 2 - 4 \rightarrow -2 \leq -2\) → True

   Since one of the inequalities is false, this option does not include (3, -2).

2. Option 2:
   \( y > -3 \)
   \( y \geq \frac{2}{3}x - 4 \)

   • For \( y > -3 \):
     \(-2 > -3\) → True
   • For \( y \geq \frac{2}{3}(3) - 4\):
     \(-2 \geq 2 - 4 \rightarrow -2 \geq -2\) → True

   Both inequalities are true, so this option includes (3, -2).

3. Option 3:
   \( y < -3 \)
   \( y \geq \frac{2}{3}x - 4 \)

   • For \( y < -3 \):
     \(-2 < -3\) → False
   • For \( y \geq \frac{2}{3}(3) - 4\):
     \(-2 \geq 2 - 4 \rightarrow -2 \geq -2\) → True

   Since one of the inequalities is false, this option does not include (3, -2).

4. Option 4:
   \( y > -2 \)
   \( y \leq \frac{2}{3}x - 4 \)

   • For \( y > -2 \):
     \(-2 > -2\) → False
   • For \( y \leq \frac{2}{3}(3) - 4\):
     \(-2 \leq 2 - 4 \rightarrow -2 \leq -2\) → True

   Since one of the inequalities is false, this option does not include (3, -2).

Conclusion:

The only system of linear inequalities that includes the point (3, -2) in its solution set is Option 2:

• \( y > -3 \)
• \( y \geq \frac{2}{3}x - 4 \)

Thus, the correct answer is Option 2.

Answer 8

Which ordered pair makes both inequalities true?y > –2x + 3y < x – 2On a coordinate plane, 2 straight lines are shown. The first solid line has a positive slope and goes through (0, negative 2) and (2, 0). Everything to the right of the line is shaded. The second dashed line has a negative slope and goes through (0, 3) and (1, 1). Everything to the right of the line is shaded. (0,0)(0,–1)(1,1)(3,0)

Answer 9

To determine which ordered pair makes both inequalities true for the given inequalities:

1. Inequalities Given:

   • \( y > -2x + 3 \)
   • \( y < x - 2 \)

2. Understanding Each Inequality:

   • For the first inequality \( y > -2x + 3 \):
     • The line \( y = -2x + 3 \) has a y-intercept of 3 and a slope of -2. The solution region is above the line (to the right, as you mentioned).
   • For the second inequality \( y < x - 2 \):
     • The line \( y = x - 2 \) has a y-intercept of -2 and a slope of 1. The solution region is below the line (also to the right, as you mentioned).

3. Visualizing the Regions:

   • To make both inequalities true, an ordered pair must lie in the intersection of the two shaded regions (i.e., it must be above the first line and below the second line).

4. Evaluating the Ordered Pairs: Now, we will check each of the provided ordered pairs against both inequalities.

   • (0, 0):

     • For \( y > -2x + 3 \): \( 0 > -2(0) + 3 \) → \( 0 > 3\) → False.
     • For \( y < x - 2 \): \( 0 < 0 - 2 \) → \( 0 < -2\) → False.
     • Not a solution.

   • (0, -1):

     • For \( y > -2x + 3 \): \( -1 > -2(0) + 3 \) → \( -1 > 3\) → False.
     • For \( y < x - 2 \): \( -1 < 0 - 2 \) → \( -1 < -2\) → False.
     • Not a solution.

   • (1, 1):

     • For \( y > -2x + 3 \): \( 1 > -2(1) + 3 \) → \( 1 > 1\) → False.
     • For \( y < x - 2 \): \( 1 < 1 - 2 \) → \( 1 < -1\) → False.
     • Not a solution.

   • (3, 0):

     • For \( y > -2x + 3 \): \( 0 > -2(3) + 3 \) → \( 0 > -6 + 3 \) → \( 0 > -3\) → True.
     • For \( y < x - 2 \): \( 0 < 3 - 2 \) → \( 0 < 1\) → True.
     • This pair satisfies both inequalities.

5. Conclusion: The ordered pair (3, 0) satisfies both inequalities \( y > -2x + 3 \) and \( y < x - 2 \).

Thus, the correct answer is (3, 0).

Answer 10

3. All values that satisfy y < 1/3x – 3 are solutions.

Answer 11

To clarify your statement, you seem to be indicating that "all values that satisfy \( y < \frac{1}{3}x - 3 \) are solutions." This phrase suggests that we are discussing a linear inequality that defines a shading region below the line described by the equation \( y = \frac{1}{3}x - 3 \) on a coordinate plane.

Context Analysis:

1. Understanding the Inequality:

   • The inequality \( y < \frac{1}{3}x - 3 \) describes all the points (x, y) that lie below the line \( y = \frac{1}{3}x - 3 \).
   • The line has a slope of \( \frac{1}{3} \) and a y-intercept at -3 (where it crosses the y-axis).

2. Shading on the Graph:

   • When graphing this inequality, the line itself is dashed (indicating that points on the line are not included in the solution set) and everything below that line is shaded to represent all points that satisfy the inequality.

3. Possible Combinations:

   • If this inequality is part of a system of inequalities, your statement would imply that all points below that line, and possibly another inequality that relates to it (if stated), would be considered solutions to the overall system.

Conclusion:

In summary, if you are indicating that all values below the line \( y < \frac{1}{3}x - 3 \) are solutions to one or more inequalities, then this is indeed true for that specific inequality. If you need further clarification or context regarding this inequality or how it fits into a larger system, feel free to provide more details!