To calculate: The solution set of the compound inequality -4\le-\frac{2}{3}p+14,

pavitorj6 2021-11-21 Answered
To calculate: The solution set of the compound inequality \(\displaystyle-{4}\le-{\frac{{{2}}}{{{3}}}}{p}+{14}\), graph it, and write it in set-builder notation and interval notation.

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Expert Answer

navratitehf
Answered 2021-11-22 Author has 297 answers
Given Information:
The compound inequality, \(\displaystyle-{4}\le-{\frac{{{2}}}{{{3}}}}{p}+{14}\)
Calculation:
The given compound inequality is \(\displaystyle-{4}\le-{\frac{{{2}}}{{{3}}}}{p}+{14}\). Subtract 14 from the inequality.
\(\displaystyle-{4}-{14}\le-{\frac{{{2}}}{{{3}}}}{p}+{14}-{14}\)
\(\displaystyle-{18}\le-{\frac{{{2}}}{{{3}}}}{p}\)
Divide the inequality by \(\displaystyle-{\frac{{{2}}}{{{3}}}}\) and change the sense of inequality,
\(\displaystyle{\frac{{-{18}}}{{{\left(-{\frac{{{2}}}{{{3}}}}\right)}}}}\geq{\frac{{-{\frac{{{2}}}{{{3}}}}}}{{{\left(-{\frac{{{2}}}{{{3}}}}\right)}}}}{p}\)
After division,
\(\displaystyle-{18}\times-{\frac{{{3}}}{{{2}}}}\geq-{\frac{{{2}}}{{{3}}}}\times-{\frac{{{3}}}{{{2}}}}{p}\)
\(\displaystyle{27}\geq{p}\)
The graph of the solution is:
image
Therefore, the solution set of compound inequality \(\displaystyle-{4}\le-{\frac{{{2}}}{{{3}}}}{p}+{14}\) is \(\displaystyle{\left\lbrace{p}{\mid}{p}\le{27}\right\rbrace}\) and interval notation is \(\displaystyle{\left(-\propto,{27}\right]}\).
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