Question

1. consider the inequality

-\\(\frac{1}{10}x + 1\frac{2}{5}\geq1\frac{7}{10}\\).
part a
solve the inequality -\\(\frac{1}{10}x + 1\frac{2}{5}\geq1\frac{7}{10}\\).
complete the sentence to describe
how to graph the solution on a
number line.
the solution can be graphed by
plotting
\\(\

$$\begin{array} { l } { \\square \\text { an open circle } } \\ { \\square \\text { a closed circle } } \\end{array}$$

\\)
at the number
\\(\

$$\begin{array} { l } { \\square - 3 } \\ { \\square 3 } \\end{array}$$

\\)
and
shading the number line to the
\\(\

$$\begin{array} { l } { \\square \\text { left } } \\ { \\square \\text { right } } \\end{array}$$

\\)
of the circle.
part b
select all the values of x that are
solutions for this inequality.
\\(\

$$\begin{array} { l l } { \\square - 8 } & { \\square 0 } \\ { \\square - 6 } & { \\square 3 } \\ { \\square - 3 } & { \\square 6 } \\end{array}$$

\\)

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(1\frac{2}{5}=\frac{1\times5 + 2}{5}=\frac{7}{5}=\frac{14}{10}\) and \(1\frac{7}{10}=\frac{1\times10+7}{10}=\frac{17}{10}\).
The inequality becomes \(-\frac{1}{10}x+\frac{14}{10}\geq\frac{17}{10}\).

Step2: Subtract \(\frac{14}{10}\) from both sides

\(-\frac{1}{10}x+\frac{14}{10}-\frac{14}{10}\geq\frac{17}{10}-\frac{14}{10}\).
We get \(-\frac{1}{10}x\geq\frac{17 - 14}{10}=\frac{3}{10}\).

Step3: Multiply both sides by \(- 10\) (and reverse the inequality sign)

When we multiply or divide an inequality by a negative number, the inequality sign flips.
\((-10)\times(-\frac{1}{10}x)\leq(-10)\times\frac{3}{10}\).
So \(x\leq - 3\).

For graphing \(x\leq - 3\) on a number - line:

• Since the inequality is \(\leq\) (not just \(<\)), we use a closed circle.
• The value is \(-3\), and we shade to the left (because \(x\) is less than or equal to \(-3\)).

For Part B:
We check each value:

• If \(x=-8\), then \(-\frac{1}{10}\times(-8)+1\frac{2}{5}=\frac{8}{10}+\frac{14}{10}=\frac{22}{10}=2.2\) and \(1\frac{7}{10}=1.7\), \(2.2\geq1.7\), so \(x = - 8\) is a solution.
• If \(x=-6\), then \(-\frac{1}{10}\times(-6)+1\frac{2}{5}=\frac{6}{10}+\frac{14}{10}=2\geq1.7\), so \(x=-6\) is a solution.
• If \(x=-3\), then \(-\frac{1}{10}\times(-3)+1\frac{2}{5}=\frac{3}{10}+\frac{14}{10}=\frac{17}{10}=1.7\geq1.7\), so \(x = - 3\) is a solution.
• If \(x = 0\), then \(-\frac{1}{10}\times0+1\frac{2}{5}=\frac{14}{10}=1.4<1.7\), so \(x = 0\) is not a solution.
• If \(x = 3\), then \(-\frac{1}{10}\times3+1\frac{2}{5}=-\frac{3}{10}+\frac{14}{10}=\frac{11}{10}=1.1<1.7\), so \(x = 3\) is not a solution.
• If \(x = 6\), then \(-\frac{1}{10}\times6+1\frac{2}{5}=-\frac{6}{10}+\frac{14}{10}=\frac{8}{10}=0.8<1.7\), so \(x = 6\) is not a solution.

Answer:

Part A:

• a closed circle
• \(-3\)
• left

Part B:

• \(-8\)
• \(-6\)
• \(-3\)