Question

1. solve the inequality and describe what its graph would look like -7 + 10z > -27

a) open circle on negative two pointed to the left b) open circle on negative thirty - four pointed to the right
c) open circle on two pointed to the left d) open circle on negative two pointed to the right

1. 10 < 3x + 19

a) x < -3 b) x < 3
c) x > -3 d) x > 3

1. which of the following inequalities is represented by the number line above?

a) 59≤4x + 15 b) 46≤1 - 5x
c) 59 ≥ 4x + 15 d) -56 ≥ 5x - 1

Explanation:

Step1: Solve -7 + 10z > -27

Add 7 to both sides:
\[

$$\begin{align*} -7+10z + 7&>-27 + 7\\ 10z&>-20 \end{align*}$$

\]
Divide both sides by 10: \(z>- 2\). The graph has an open - circle on negative two pointed to the right.

Step2: Solve 10 < 3x + 19

Subtract 19 from both sides:
\[

$$\begin{align*} 10-19&<3x+19 - 19\\ -9&<3x \end{align*}$$

\]
Divide both sides by 3: \(x>-3\).

Step3: Analyze the number - line

The number - line has a closed circle at 11 and the line is pointed to the right, which means \(x\geq11\).
For option a) \(59\leq4x + 15\), subtract 15 from both sides: \(59-15\leq4x+15 - 15\), \(44\leq4x\), divide by 4, \(x\geq11\).
For option b) \(46\leq1 - 5x\), subtract 1 from both sides: \(46 - 1\leq1-5x - 1\), \(45\leq - 5x\), divide by - 5 and reverse the inequality sign: \(x\leq - 9\).
For option c) \(59\geq4x + 15\), subtract 15 from both sides: \(59-15\geq4x+15 - 15\), \(44\geq4x\), \(x\leq11\).
For option d) \(-56\geq5x - 1\), add 1 to both sides: \(-56 + 1\geq5x-1 + 1\), \(-55\geq5x\), \(x\leq - 11\).

Answer:

1. d) Open circle on negative two pointed to the right
2. c) \(x > - 3\)
3. a) \(59\leq4x + 15\)