Question 1

Graph the system of inequalities.
-$\begin{array}{l}
y>-5 \
x \geq 3
\end{array}$

Accepted Answer

The answer of Graph the system of inequalities.
-\(\begin{array}{l}
y>-5 \
x \geq...

Question 2

Graph the inequality.
-$$y \leq x ^ { 2 } + 1$$

Accepted Answer

The answer of Graph the inequality.
-\[y \leq x ^ {...

Question 3

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x y = 56 \
x + y = - 15
\end{array}$$
(i) $( - 8 , - 7 )$; (ii) $( 7,8 )$

Accepted Answer

For (i) $(-8, -7)$, substituting into the equations gives: $(-8) \times (-7) = 56$ and $(-8) + (-7) = -15$, which satisfies both equations. For (ii) $(7, 8)$, substituting gives: $7 \times 8 = 56$ but $7 + 8 = 15$, which does not satisfy the second equation.

Question 4

Determine if the ordered pair is a solution to the system of linear inequalities in two variables.
-$$\begin{array} { l } 
2 x + 5 y < 14 \
- 5 x + 2 y > 14 \
x < 1 \
y \geq - 4
\end{array}$$
$( 1 , - 4 )$

Accepted Answer

Substituting $x = 1$ and $y = -4$ into each inequality:1. $2(1) + 5(-4) < 14$ simplifies to $2 - 20 < 14$, which is $-18 < 14$, true.2. $-5(1) + 2(-4) > 14$ simplifies to $-5 - 8 > 14$, which is $-13 > 14$, false.3. $1 < 1$ is false because the inequality strictly says "less than".4. $-4 \geq -4$ is true.Since not all inequalities are satisfied, the ordered pair $(1, -4)$ is not a solution to the system.

Question 5

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
3 x - y = - 17 \
y = x ^ { 2 } + 7
\end{array}$$
(i) $( 5,32 )$; (ii) $( - 2,11 )$

Accepted Answer

For (i) $ (5,32) $, substituting $ x = 5 $ into the equations gives $ 3(5) - y = -17 $ and $ y = 5^2 + 7 $. The first equation simplifies to $ 15 - y = -17 $, which does not directly relate to the solution, but checking the second equation, $ y = 25 + 7 = 32 $, which matches, indicating the first equation was incorrectly assessed. Correctly, $ 3(5) - 32 = 15 - 32 = -17 $, so both conditions are satisfied. For (ii) $ (-2,11) $, substituting $ x = -2 $ gives $ 3(-2) - y = -17 $ and $ y = (-2)^2 + 7 $. The first equation simplifies to $ -6 - y = -17 $, and the second to $ y = 4 + 7 = 11 $. For the first equation, correctly solving for $ y $ gives $ -6 - y = -17 $ or $ y = 11 $, which matches the given $ y $ value. Thus, both ordered pairs satisfy the given system of inequalities.

Question 6

Graph the system of inequalities.
-$\begin{array}{l}
y<-x+3 \
y>2 x-3
\end{array}$

Accepted Answer

The answer of Graph the system of inequalities.
-$\begin{array}{l}
y<-x+3 \
y>2 x-3
\end{array}$

...

Question 7

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x y = 54 \
x ^ { 2 } + y ^ { 2 } = 117
\end{array}$$
(i) $( 6 , - 9 )$; (ii) $( - 9 , - 6 )$

Accepted Answer

For (i) $6 \times -9 = -54$, not 54, and $6^2 + (-9)^2 = 36 + 81 = 117$, so only the second equation is satisfied. For (ii) $(-9) \times (-6) = 54$, and $(-9)^2 + (-6)^2 = 81 + 36 = 117$, so both equations are satisfied.

Question 8

Graph the system of inequalities.
-$$4 x - y \leq 4$$
$$x + 2 y \geq - 8$$

Accepted Answer

The answer of Graph the system of inequalities.
-\[4 x -...

Question 9

Determine if the ordered pair is a solution to the system of linear inequalities in two variables.
-$$\begin{array} { l } 
x + y \geq 2 \
- 3 x + y \leq 11 \
x \geq 0 \
y \geq 1
\end{array}$$

Accepted Answer

To determine if an ordered pair is a solution to the system of linear inequalities, it must satisfy all the given inequalities. Without a specific ordered pair provided in the question, we can't directly evaluate, but the question seems to imply checking if a general solution exists that meets all conditions. Since the system of inequalities defines a region in the coordinate plane where all conditions can simultaneously be true (for example, a point in the first quadrant that satisfies all linear inequalities), there exists at least one ordered pair that satisfies the system, hence the answer is "Yes."

Question 10

Graph the system of inequalities.
-$\begin{array}{l}
3 x+y>6 \
3 x+y<1
\end{array}$

Accepted Answer

The answer of Graph the system of inequalities.
-\(\begin{array}{l}
3 x+y>6 \
3...

Question 11

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 61 \
x + y = - 11
\end{array}$$
(i) $( 5,6 )$; (ii) $( 6,5 )$

Accepted Answer

For (i) $5^2 + 6^2 = 25 + 36 = 61$ and $5 + 6 = 11$, which does not satisfy the second equation since it should be $-11$. For (ii) $6^2 + 5^2 = 36 + 25 = 61$ and $6 + 5 = 11$, which also does not satisfy the second equation.

Question 12

Determine if the ordered pair is a solution to the system of linear inequalities in two variables.
-$$\begin{array} { l } 
x + y \geq 4 \
- 3 x + y \leq 12
\end{array}$$

Accepted Answer

The ordered pair satisfies both inequalities, thus it is a solution to the system.

Question 13

Graph the inequality.
-$$x ^ { 2 } + y ^ { 2 } > 4$$

Accepted Answer

The answer of Graph the inequality.
-\[x ^ { 2 }...

Question 14

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x + y = 13 \
y = x ^ { 2 } - 14 x + 49
\end{array}$$
(i) $( - 4,17 )$; (ii) $( 9,4 )$

Accepted Answer

To determine if each ordered pair is a solution to the given system of inequalities, we substitute the x and y values from each ordered pair into both equations and check if they satisfy both equations.For (i) $(-4, 17)$:1. $x + y = 13$ becomes $(-4) + 17 = 13$, which is true.2. $y = x^2 - 14x + 49$ becomes $17 = (-4)^2 - 14(-4) + 49$, which simplifies to $17 = 16 + 56 + 49$, which is not true. So, (i) is not a solution.For (ii) $(9, 4)$:1. $x + y = 13$ becomes $9 + 4 = 13$, which is true.2. $y = x^2 - 14x + 49$ becomes $4 = 9^2 - 14(9) + 49$, which simplifies to $4 = 81 - 126 + 49$, which is true. So, (ii) is a solution.Therefore, the correct answer is (i) No; (ii) Yes.

Question 15

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x ^ { 2 } = y ^ { 2 } + 39 \
x - y = 3
\end{array}$$
(i) $( 8 , - 5 )$; (ii) $( - 8 , - 5 )$

Accepted Answer

The answer of Determine if each ordered pair is a...

Question 16

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
x y = 1 \
- 7 x - y = - 8
\end{array}$$
(i) $\left( \frac { 1 } { 7 } , 7 \right)$;(ii) $( 1,1 )$

Accepted Answer

Both pairs satisfy the equations: (i) $\frac{1}{7} \times 7 = 1$ and $-7 \times \frac{1}{7} - 7 = -8$; (ii) $1 \times 1 = 1$ and $-7 \times 1 - 1 = -8$.

Question 17

Determine if the ordered pair is a solution to the system of linear inequalities in two variables.
-$$\begin{array} { l } 
3 x + 3 y < 11 \
- 3 x + 3 y > 11
\end{array}$$

Accepted Answer

To determine if an ordered pair is a solution to a system of inequalities, we substitute the values of $x$ and $y$ into each inequality. However, without the specific ordered pair provided in the question, we cannot directly verify if it satisfies both inequalities. The answer provided as "B) No" suggests a misunderstanding or a missing part of the question, as we need an ordered pair to make a determination.

Question 18

Graph the inequality.
-

Accepted Answer

The answer of Graph the inequality.
- 
...

Question 19

Determine if each ordered pair is a solution to the given system of inequalities in two variables.
-$$\begin{array} { l } 
y = x - 1 \
y ^ { 2 } = - 4 x
\end{array}$$
(i) $( - 1 , - 2 )$; (ii) $( 1,0 )$

Accepted Answer

For (i) $(-1, -2)$, substituting $x = -1$ into $y = x - 1$ gives $y = -2$, which matches. Substituting $x = -1$ into $y^2 = -4x$ gives $y^2 = 4$, and since $y = -2$, $y^2 = 4$ is true. Thus, (i) is a solution.For (ii) $(1, 0)$, substituting $x = 1$ into $y = x - 1$ gives $y = 0$, which matches. However, substituting $x = 1$ into $y^2 = -4x$ gives $y^2 = -4$, which is not possible since $y^2$ cannot be negative for real numbers. Thus, (ii) is not a solution.

Question 20

Graph the system of inequalities.
-$\begin{array}{l}
-3 x+y<6 \
-3 x+y>1
\end{array}$

Accepted Answer

The answer of Graph the system of inequalities.
-\(\begin{array}{l}
-3 x+y<6 \
-3...

Quiz 7: Systems of Equations and Inequalities

Graph the system of inequalities. - $\begin{array}{l}y>-5 \\x \geq 3\end{array}$ $Graph the system of inequalities. - \begin{array}{l} y>-5 \\ x \geq 3 \end{array} A) B) C) D)$

Graph the inequality. - $y \leq x ^ { 2 } + 1$ $Graph the inequality. - y \leq x ^ { 2 } + 1 A) B) C) D)$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x y = 56 \\x + y = - 15\end{array}$ (i) $( - 8 , - 7 )$ ; (ii) $( 7,8 )$

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - $\begin{array} { l } 2 x + 5 y < 14 \\- 5 x + 2 y > 14 \\x < 1 \\y \geq - 4\end{array}$ $( 1 , - 4 )$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } 3 x - y = - 17 \\y = x ^ { 2 } + 7\end{array}$ (i) $( 5,32 )$ ; (ii) $( - 2,11 )$

Graph the system of inequalities. - $\begin{array}{l}y<-x+3 \\y>2 x-3\end{array}$ $Graph the system of inequalities. - \begin{array}{l} y<-x+3 \\ y>2 x-3 \end{array} A) B) C) D)$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x y = 54 \\x ^ { 2 } + y ^ { 2 } = 117\end{array}$ (i) $( 6 , - 9 )$ ; (ii) $( - 9 , - 6 )$

Graph the system of inequalities. - $4 x - y \leq 4$ $x + 2 y \geq - 8$ $Graph the system of inequalities. - 4 x - y \leq 4 x + 2 y \geq - 8 A) B) C) D)$

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - $\begin{array} { l } x + y \geq 2 \\- 3 x + y \leq 11 \\x \geq 0 \\y \geq 1\end{array}$

Graph the system of inequalities. - $\begin{array}{l}3 x+y>6 \\3 x+y<1\end{array}$ $Graph the system of inequalities. - \begin{array}{l} 3 x+y>6 \\ 3 x+y<1 \end{array} A) No solution B) C) D)$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x ^ { 2 } + y ^ { 2 } = 61 \\x + y = - 11\end{array}$ (i) $( 5,6 )$ ; (ii) $( 6,5 )$

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - $\begin{array} { l } x + y \geq 4 \\- 3 x + y \leq 12\end{array}$

Graph the inequality. - $x ^ { 2 } + y ^ { 2 } > 4$ $Graph the inequality. - x ^ { 2 } + y ^ { 2 } > 4 A) B) C) D)$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x + y = 13 \\y = x ^ { 2 } - 14 x + 49\end{array}$ (i) $( - 4,17 )$ ; (ii) $( 9,4 )$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x ^ { 2 } = y ^ { 2 } + 39 \\x - y = 3\end{array}$ (i) $( 8 , - 5 )$ ; (ii) $( - 8 , - 5 )$

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } x y = 1 \\- 7 x - y = - 8\end{array}$ (i) $\left( \frac { 1 } { 7 } , 7 \right)$ ;(ii) $( 1,1 )$

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - $\begin{array} { l } 3 x + 3 y < 11 \\- 3 x + 3 y > 11\end{array}$

Graph the inequality. -

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - $\begin{array} { l } y = x - 1 \\y ^ { 2 } = - 4 x\end{array}$ (i) $( - 1 , - 2 )$ ; (ii) $( 1,0 )$

Graph the system of inequalities. - $\begin{array}{l}-3 x+y<6 \\-3 x+y>1\end{array}$ $Graph the system of inequalities. - \begin{array}{l} -3 x+y<6 \\ -3 x+y>1 \end{array} A) B) No solution C) D)$

Quiz 7: Systems of Equations and Inequalities

Graph the system of inequalities. - y>−5x≥3\begin{array}{l}y>-5 \\x \geq 3\end{array}y>−5x≥3​

Graph the inequality. - y≤x2+1y \leq x ^ { 2 } + 1y≤x2+1

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - xy=56x+y=−15\begin{array} { l } x y = 56 \\x + y = - 15\end{array}xy=56x+y=−15​ (i) (−8,−7) ( - 8 , - 7 ) (−8,−7) ; (ii) (7,8) ( 7,8 ) (7,8)

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - 2x+5y<14−5x+2y>14x<1y≥−4\begin{array} { l } 2 x + 5 y < 14 \\- 5 x + 2 y > 14 \\x < 1 \\y \geq - 4\end{array}2x+5y<14−5x+2y>14x<1y≥−4​ (1,−4) ( 1 , - 4 ) (1,−4)

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - 3x−y=−17y=x2+7\begin{array} { l } 3 x - y = - 17 \\y = x ^ { 2 } + 7\end{array}3x−y=−17y=x2+7​ (i) (5,32) ( 5,32 ) (5,32) ; (ii) (−2,11) ( - 2,11 ) (−2,11)

Graph the system of inequalities. - y<−x+3y>2x−3\begin{array}{l}y<-x+3 \\y>2 x-3\end{array}y<−x+3y>2x−3​

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - xy=54x2+y2=117\begin{array} { l } x y = 54 \\x ^ { 2 } + y ^ { 2 } = 117\end{array}xy=54x2+y2=117​ (i) (6,−9) ( 6 , - 9 ) (6,−9) ; (ii) (−9,−6) ( - 9 , - 6 ) (−9,−6)

Graph the system of inequalities. - 4x−y≤44 x - y \leq 44x−y≤4 x+2y≥−8x + 2 y \geq - 8x+2y≥−8

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - x+y≥2−3x+y≤11x≥0y≥1\begin{array} { l } x + y \geq 2 \\- 3 x + y \leq 11 \\x \geq 0 \\y \geq 1\end{array}x+y≥2−3x+y≤11x≥0y≥1​

Graph the system of inequalities. - 3x+y>63x+y<1\begin{array}{l}3 x+y>6 \\3 x+y<1\end{array}3x+y>63x+y<1​

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - x2+y2=61x+y=−11\begin{array} { l } x ^ { 2 } + y ^ { 2 } = 61 \\x + y = - 11\end{array}x2+y2=61x+y=−11​ (i) (5,6) ( 5,6 ) (5,6) ; (ii) (6,5) ( 6,5 ) (6,5)

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - x+y≥4−3x+y≤12\begin{array} { l } x + y \geq 4 \\- 3 x + y \leq 12\end{array}x+y≥4−3x+y≤12​

Graph the inequality. - x2+y2>4x ^ { 2 } + y ^ { 2 } > 4x2+y2>4

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - x+y=13y=x2−14x+49\begin{array} { l } x + y = 13 \\y = x ^ { 2 } - 14 x + 49\end{array}x+y=13y=x2−14x+49​ (i) (−4,17) ( - 4,17 ) (−4,17) ; (ii) (9,4) ( 9,4 ) (9,4)

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - x2=y2+39x−y=3\begin{array} { l } x ^ { 2 } = y ^ { 2 } + 39 \\x - y = 3\end{array}x2=y2+39x−y=3​ (i) (8,−5) ( 8 , - 5 ) (8,−5) ; (ii) (−8,−5) ( - 8 , - 5 ) (−8,−5)

Determine if each ordered pair is a solution to the given system of inequalities in two variables. - xy=1−7x−y=−8\begin{array} { l } x y = 1 \\- 7 x - y = - 8\end{array}xy=1−7x−y=−8​ (i) (17,7) \left( \frac { 1 } { 7 } , 7 \right) (71​,7) ;(ii) (1,1) ( 1,1 ) (1,1)

Determine if the ordered pair is a solution to the system of linear inequalities in two variables. - 3x+3y<11−3x+3y>11\begin{array} { l } 3 x + 3 y < 11 \\- 3 x + 3 y > 11\end{array}3x+3y<11−3x+3y>11​