Question

Which statements are true about the graph of y s 3x + 1 and y 2 -x + 2? Check all that apply.

O The slope of one boundary line is 2.
O Both boundary lines are solid.
A solution to the system is (1, 3).
O Both inequalities are shaded below the boundary lines
O The boundary lines intersect.

Answer 1

3 Answers:

• B) Both boundary lines are solid.
• C) A solution to the system is (1,3).
• E) The boundary lines intersect.

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Step-by-step explanation:

Let's go through each of the answer choices to see which are true and which are false.

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Choice A

The general slope intercept form is y = mx+b

m = slope and b = y intercept.

For y = 3x+1, the boundary line of the first inequality, the slope is 3.

For y = -x+2, the slope is -1.

None of the slopes are 2.

Choice A is false.

We'll use the concept of slope again in choice E.

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Choice B

The "or equal to" part of the inequality sign is what directly determines the boundary line being solid. This is because we include points on the boundary.

Choice B is true.

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Choice C

Plug x = 1 and y = 3 into the first inequality

`y \le 3x+1\\\\3 \le 3(1)+1\\\\3 \le 3+1\\\\3 \le 4\\\\`

The last inequality is true, so the first inequality is true when (x,y) = (1,3). This makes (1,3) a solution to
`y \le 3x+1\\\\`

Repeat those steps for the other inequality given to us

`y \ge -x+2\\\\3 \ge -1+2\\\\3 \ge 1\\\\`

which is also true, so that makes (1,3) also a solution to
`y \ge -x+2\\\\`

The point (1,3) is a solution to both inequalities at the same time, making it a solution to the system overall.

Choice C is true

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Choice D

When y is fully isolated, the "less than" inequality indicates the shading is below the boundary. This is due to us considering points of y coordinates smaller than the boundary line. So we'll shade below the boundary for
`y \le 3x+1\\\\`

We don't do the same for
`y \ge -x+2\\\\`. Instead, we'll shade above the solid boundary line.

Choice D is false.

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Choice E

If two lines have equal slopes, but different y intercepts, then we have parallel lines. Parallel lines never intersect.

If on the other hand, the slopes are different values, then the two lines will intersect somewhere on the xy grid. There will only be exactly one point of intersection.

The given system has boundary lines of slopes 3 and -1, for the first and second inequality respectively. Refer back to choice A. These differing slopes tell us that the boundary lines intersect somewhere.

Choice E is true

Answer 2

Let's go through all the options one at a time and look for the correct ones

Option 1: The slope of one boundary line is 2

We have 2 equations of lines, where the coefficients of are 3 and -1 respectively

because the coefficient of x denotes the slope of a line, we know that the lines have the slope 3 and -1, not 2

Hence, this option is Incorrect

Option 2: Both boundary lines are solid

In order for the boundary lines to be solid, the inequality must have an 'equal to', like ≤ (less than or equal to) or ≥ (greater than or equal to)

we can see that that's the case in our case and hence, this option is Correct

Option 3: A solution to the system is (1, 3)

To confirm this, we'll plug these coordinates into the given inequalities and see if it stands correct

y ≤ 3x + 1

3 ≤ 3(1) + 1

3 ≤ 4 which is correct because 3 is less than 4

Second equation:

y ≥ 2 - x

3 ≥ 2 - 1

3 ≥ 1

Which is also true because 3 is greater than 1

Now, we can say that (1 , 3) is a solution to the system because it satisfies both the equations and is Correct

Option 4: Both inequalities are shaded below the boundary lines

For an inequality to be shaded below the boundary line, it must have the ≤ inequality (in case of solid line) and < inequality (in case of dotted line)

because the second inequality listed includes the ≥ inequality, which was not mentioned above, it won't be shaded below

another way to think about it is that any 'greater than' inequality will shade everything above the line and the 'lesser than' inequality will shade below the line

which means that this option is Incorrect

Option 5: The boundary lines intersect

In order for the boundary lines to intersect, they must have have different slopes.

as we mentioned in the explanation of the first option, that the slopes of the lines is 3 and -1, which are different slopes

Therefore, this option is Correct