7.11: Proficiency Exam

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Proficiency Exam

For the following problems, construct a coordinate system and graph the inequality.

Exercise $\PageIndex{1}$

$−6x+4>−14$

Answer

$x<3$

$A number line labeled x with arrows on each end, labeled from negative three to four, in increments of one. There is an open circle at three. A dark line is orginating from this circle and heading towards the left of three.$

Exercise $\PageIndex{2}$

$−8<x+6≤−4$

Answer

$−14<x≤−10$

$A number line labeled x with arrows on each end, labeled at negative fourteen and negative ten. There is a closed circle at negative ten and an open circle at negative fourteen. These circles are connected by a black line$

Exercise $\PageIndex{3}$

Plot the ordered pairs $(3, 1),(−2, 4),(0, 5),(−2, −2)$.

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer: $Total four points plotted in an xy-coordinate plane. The coordinates of these points are negative two, negative two; negative two, four; zero, five and three, one.$

Exercise $\PageIndex{4}$

As accurately as possible, label the coordinates of the points that have been plotted on the graph.

$Total four points plotted in an xy plane. The coordinates of these points are negative three, zero; zero, one; three, three and two, negative three.$

Answer: $(0,1),(3,3),(−3,0),(2,−3)$

Exercise $\PageIndex{5}$

What is the geometric structure of the graph of all the solutions to the equation $2y+3x=−4$?

Answer: straight line

Exercise $\PageIndex{6}$

In what form is the linear equation in two variables $ax+by=c$?

Answer: general form

Exercise $\PageIndex{7}$

In what form is the linear equation in two variables $y=mx+b$?

Answer: slope-intercept

Exercise $\PageIndex{8}$

If an ordered pair is a solution to a linear equation in two variables, where does it lie geometrically?

Answer: It lies on the line.

Exercise $\PageIndex{9}$

Consider the graph of $y=\dfrac{2}{7}x+16$. If we were to place our pencil at any point on the line and then move it horizontally $7$ units to the right, how many units and in what direction would we have to move our pencil to get back on the line?

Answer: $2$ units up

For the following problems, find the slope, if it exists, of the line containing the following points.

Exercise $\PageIndex{10}$

(−6, −1) and (0, 8)

Answer: $\dfrac{3}{2}$

Exercise $\PageIndex{11}$

(−2, −8) and (−2, 10)

Answer: no slope; vertical line at $x=−2$

Exercise $\PageIndex{12}$

Determine the slope and $y$−intercept of the line $3y+2x+1=0$.

Answer: slope = $-\dfrac{2}{3}$, $y$-intercept is $(0, -\dfrac{1}{3})$

Exercise $\PageIndex{13}$

As we look at a graph left to right, do lines with a positive slope rise or decline?

Answer: rise

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form

Exercise $\PageIndex{14}$

Slope = $4$, $y$-intercept = $−3$.

Answer: $y=4x−3$

Exercise $\PageIndex{15}$

Slope = $-\dfrac{3}{2}$, $y$-intercept = $\dfrac{4}{3}$.

Answer: $y = -\dfrac{3}{2}x + \dfrac{4}{3}$

Exercise $\PageIndex{16}$

Slope = $\dfrac{2}{3}$, passes through $(-1, 2)$

Answer: $y = \dfrac{2}{3}x + \dfrac{8}{3}$

Exercise $\PageIndex{17}$

Slope = $7$, passes through $(0, 0)$

Answer: $y=7x$

Exercise $\PageIndex{18}$

Passes through the points $(5, 2)$ and $(2, 1)$.

Answer: $y = \dfrac{1}{3}x + \dfrac{1}{3}$

For the following problems, graph the equation of inequality.

Exercise $\PageIndex{19}$

$y = \dfrac{1}{3}x - 2$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer

$y = \dfrac{1}{3}x - 2$

$A graph of a line passing through two points with coordinates zero, negative two and three, negative one.$

Exercise $\PageIndex{20}$

$5y−2x+15=0$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer: $A graph of a line passing through two points with coordinates zero, negative three and five, negative one.$

Exercise $\PageIndex{21}$

$4(x+y)=8$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer

$4(x+y)=8$

$A graph of a line passing through four points with coordinates negative two, four; zero, two; two, zero; and four, negative two.$

Exercise $\PageIndex{22}$

$\dfrac{3}{2}y + 2 = 0$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer: $A graph of a line parallel to x-axis and passing through a point with coordinates zero, negative four over three.$

Exercise $\PageIndex{23}$

$x=−2$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer

$x=−2$

$A graph of a line parallel to y-axis in an xy plane. The line crosses the x-axis at x equals negative two.$

Exercise $\PageIndex{24}$

$2x+3y>6$

$An xy-plane with gridlines, labeled negative five and five on the both axes.$

Answer: $A dashed line in an xy plane passing through two points with coordinates zero, two and three, zero. The region above the line is shaded.$

Exercise $\PageIndex{25}$

Reading only from the graph, determine the equation of the line.

$A graph of a line sloped up and to the left.$

Answer: $y = -\dfrac{1}{3}x + 3$

Search

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