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Math Problems

Grade 8

Multiply and divide rational numbers

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Q. ![Linear inequality word problems foundations](https://externalquestionsimg.blob.core.windows.net/external-question-images/images/images/ka/DigitalSAT/Foundations%

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Algebra/Linear%

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inequality%

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word%

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problems%

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foundations/Q

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Identify variables and inequality: Identify the variables and the inequality from the problem. $\newline$ Pat earns $\$12$ per hour and has already worked for $5$ hours. Pat needs to earn at least $\$140$ in total. Let's denote the number of additional hours Pat needs to work as $h$ . $\newline$ The inequality representing the situation is: $\newline$ $12 \times 5 + 12 \times h \geq 140$
Simplify inequality by multiplying: Simplify the inequality by multiplying the known values. $\newline$ Calculate the earnings from the $5$ hours already worked: $\newline$ $12 \times 5 = 60$ $\newline$ So the inequality becomes: $\newline$ $60 + 12h \geq 140$
Isolate variable: Isolate the variable ＂h＂ on one side of the inequality. $\newline$ Subtract $60$ from both sides of the inequality to isolate the term with ＂h＂: $\newline$ $60 + 12h - 60 \geq 140 - 60$ $\newline$ This simplifies to: $\newline$ $12h \geq 80$
Solve for h: Solve for ＂h＂ by dividing both sides of the inequality by $12$ . $\newline$ Divide both sides by $12$ to find the minimum number of hours ＂h＂: $\newline$ $\frac{12h}{12} \geq \frac{80}{12}$ $\newline$ This simplifies to: $\newline$ $h \geq \frac{80}{12}$
Calculate exact value: Calculate the exact value or the smallest integer value greater than or equal to the result. $\newline$ Calculate the value of $h$ : $\newline$ $\frac{80}{12} = 6.666...$ $\newline$ Since Pat cannot work a fraction of an hour, we round up to the nearest whole hour. Therefore, Pat must work at least $7$ hours.