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phrase I. The difference between three times \( x \) and fifteen is greater than or equal to five 2. Five more than sixteen times \( x \) is less than or equal to six 3. Three more than two times \( x \) is less than seven \( \square \) 4. Five less than four times \( x \) is less than or equal to sixteen 5. Six times the sum of \( x \) and twelve is less than fourteen 6. The difference between fifteen and two times \( x \) is greater than five 7. The difference between eleven and four times \( x \) is greater than or equal to three 8. The sum of negative three times \( x \) and five is less than or equal to negative four 9. Fourteen less than five times \( x \) is at most eleven \( \qquad \) 10. Twice the sum of nine and \( x \) is greater than twenty II. Ten less than three times \( x \) is greater than eleven 12. Thirteen plus five times \( x \) is no more than thirty 13. Thirteen more than three times \( x \) is no more than the opposite of eleven 14. Half of the sum of \( x \) and six is no less than twenty 15. The difference between negative five times \( x \) and eight is greater than twelve. Solve only your inequalities! Look for your answer at the bottom. \[ \begin{array}{ll} N \quad 2 x+3 \leq 7 & E \\ C & 14-5 x \leq 11 \\ \text { C } 15-2 x>5 & \text { R } \\ F(9+x)>20 \\ E \quad 1 / 2 x+6 x \leq 30 & \text { D } \end{array} 6(x+12)<141 \] \[ \text { L } 5 x-14 \leq 11 \quad H \quad-3 x-5<-4 \] \[ \text { U } 3 x-15 \geq 5 \quad \text { A } 1 / 2(x+6) \geq 20 \] \[ E \quad 6(x-12)>14 \backslash \text { H } \quad 11-4 x \geq 3 \] \[ 3 x-10>11 \quad 0 \quad-5 x-8>12 \] \[ \vee 16 x+5<6 \quad \& \quad 3 x+13 \leq-11 \] \[ \text { Y } 4 x-5 \geq 16 \quad \text { \& } 16 x+5 \leq 6 \]

Ask by Bob Mejia. in the United States
Feb 27,2025

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Here are the solutions to the inequalities: 1. \( x \leq 2 \) 2. \( x \geq \frac{3}{5} \) 3. \( x < 5 \) 4. \( x < 11.5 \) 5. \( x \leq 5 \) 6. \( x > -\frac{1}{3} \) 7. \( x \geq \frac{20}{3} \) 8. \( x \geq 34 \) 9. \( x > \frac{43}{3} \) 10. \( x \leq 2 \) 11. \( x > 7 \) 12. \( x < -4 \) 13. \( x < \frac{1}{16} \) 14. \( x \leq -8 \) 15. \( x \geq \frac{21}{4} \)

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Beyond the Answer

When it comes to solving inequalities, understanding the historical progression of mathematical concepts can be quite fascinating! Inequalities have roots tracing back to ancient civilizations, including the Greeks who initially tackled the idea of comparisons in their geometric proofs. Over time, the formal notation we use today was developed in the 16th century by mathematician René Descartes, paving the way for algebraic expressions to flourish in modern mathematics! If you ever find yourself grappling with inequalities, don’t fret! A common mistake is flipping the inequality sign when multiplying or dividing by a negative number. This little detail trips up even the most seasoned problem-solvers. Remember: Always pay attention to your signs, and you’ll stay on the right track. Happy solving!

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