Question

circle the letter of the correct answer.

1. this number line shows the solution of each inequality except

a. $x\geq -3$.
b. $x - 3\geq -6$.
c. $x + 3\geq 0$.
d. $-x\geq 6$.
e. $\frac{x}{3}\geq -1$.

1. the solution of $-7q + 3\lt -11$ is

a. $q\lt 2$.
b. $q\gt 2$.
c. $q\gt 7$.
d. $q\lt -7$.
e. $q\lt 1$.

1. the profit from selling vitamin x is $5.00 a bottle. the profit from selling vitamin y is $6.00 a bottle. which of the following combinations of sales gives a profit of at least $42.00?

a. 5 bottles of x and 2 bottles of y
b. 1 bottle of x and 5 bottles of y
c. 4 bottles of x and 4 bottles of y
d. 6 bottles of y (none of x)
e. 7 bottles of brand x (none of y)

1. which of the following points is a solution of $y\leq 5x - 1$?

a. $(0, 2)$
b. $(-3, 0)$
c. $(-1, -5)$
d. $(2, 11)$
e. $(7, 2)$

1. which of the following points is not a solution of $y\leq x$?

a. $(5, -3)$
b. $(-1, -1)$
c. $(-6, 4)$
d. $(2, -6)$
e. $(3, 1)$

1. for which inequality does the graph show the solution?

a. $y\gt -x + 1$
b. $y\lt -x + 1$
c. $y\gt x + 1$
d. $y\lt x + 1$
e. $y\gt x$

Explanation:

1.

Step1: Solve each inequality

• For \(A\): \(x\geq - 3\)
• For \(B\): \(x-3\geq - 6\), then \(x\geq - 6 + 3=-3\)
• For \(C\): \(x + 3\geq0\), then \(x\geq - 3\)
• For \(D\): \(-x\geq6\), then \(x\leq - 6\)
• For \(E\): \(\frac{x}{3}\geq - 1\), then \(x\geq - 3\)

2.

Step1: Solve the inequality \(-7q + 3\lt - 11\)

Subtract \(3\) from both sides: \(-7q\lt - 11 - 3=-14\)
Divide both sides by \(-7\) (and reverse the inequality sign): \(q\gt2\)

3.

Step1: Calculate profit for each option

• Option \(A\): \(5\times5+2\times6 = 25 + 12=37\lt42\)
• Option \(B\): \(1\times5+5\times6=5 + 30 = 35\lt42\)
• Option \(C\): \(4\times5+4\times6=20 + 24=44\geq42\)
• Option \(D\): \(6\times6=36\lt42\)
• Option \(E\): \(7\times5=35\lt42\)

4.

Step1: Substitute each point into \(y\leq5x - 1\)

• For \(A(0,2)\): \(2\leq5\times0 - 1=-1\) (False)
• For \(B(-3,0)\): \(0\leq5\times(-3)-1=-16\) (False)
• For \(C(-1,-5)\): \(-5\leq5\times(-1)-1=-6\) (False)
• For \(D(2,11)\): \(11\leq5\times2 - 1 = 9\) (False)
• For \(E(7,2)\): \(2\leq5\times7-1=34\) (True)

5.

Step1: Substitute each point into \(y\leq x\)

• For \(A(5,-3)\): \(-3\leq5\) (True)
• For \(B(-1,-1)\): \(-1\leq - 1\) (True)
• For \(C(-6,4)\): \(4\leq - 6\) (False)
• For \(D(2,-6)\): \(-6\leq2\) (True)
• For \(E(3,1)\): \(1\leq3\) (True)

6.

Step1: Analyze the boundary line and shading

The boundary line has a slope of \(-1\) and \(y\)-intercept of \(1\), equation \(y=-x + 1\). The shading is above the line (dashed line, so strict inequality), so \(y\gt - x + 1\)

Answer:

1. D
2. B
3. C
4. E
5. C
6. A