a) Solve [tex]\frac{4}{5} x-(x+3)<\frac{1}{3}(x-1)[/tex]
b) If [tex]\frac{3}{2 y-\frac{1}{2}}=\frac{\frac{1}{3}}{\frac{1}{4} y+1}[/tex], find [tex]y[/tex]
Answer (2)
Solve the inequality by isolating x : -5"> 5 4 x − ( x + 3 ) < 3 1 ( x − 1 ) ⇒ x > − 5 .
Solve the equation by cross-multiplication and isolating y : 2 y − 2 1 3 = 4 1 y + 1 3 1 ⇒ y = − 38 .
The solution to the inequality is -5"> x > − 5 .
The solution to the equation is y = − 38 , so the final answer is − 38 .
Explanation
Problem Introduction We are given two problems: a) Solve the inequality 5 4 x − ( x + 3 ) < 3 1 ( x − 1 ) .
b) Solve the equation 2 y − 2 1 3 = 4 1 y + 1 3 1 for y .
Solving the Inequality a) First, let's solve the inequality. We have 5 4 x − ( x + 3 ) < 3 1 ( x − 1 ) 5 4 x − x − 3 < 3 1 x − 3 1 Combining the x terms on the left side, we get ( 5 4 − 1 ) x − 3 < 3 1 x − 3 1 − 5 1 x − 3 < 3 1 x − 3 1 Now, we want to isolate x . Add 5 1 x to both sides and add 3 1 to both sides: − 3 + 3 1 < 3 1 x + 5 1 x − 3 8 < ( 3 1 + 5 1 ) x − 3 8 < 15 8 x Now, divide both sides by 15 8 . Since 15 8 is positive, the inequality sign does not change: 15 8 − 3 8 < x − 3 8 ⋅ 8 15 < x − 5 < x So, -5"> x > − 5 .
Solving the Equation b) Now, let's solve the equation. We have 2 y − 2 1 3 = 4 1 y + 1 3 1 Cross-multiply to get rid of the fractions: 3 ( 4 1 y + 1 ) = 3 1 ( 2 y − 2 1 ) 4 3 y + 3 = 3 2 y − 6 1 Now, we want to isolate y . Subtract 4 3 y from both sides and add 6 1 to both sides: 3 + 6 1 = 3 2 y − 4 3 y 6 19 = ( 3 2 − 4 3 ) y 6 19 = ( 12 8 − 12 9 ) y 6 19 = − 12 1 y Multiply both sides by − 12 :
6 19 ⋅ ( − 12 ) = y 19 ⋅ ( − 2 ) = y − 38 = y So, y = − 38 .
Final Answer Therefore, the solution to the inequality is -5"> x > − 5 , and the solution to the equation is y = − 38 .
Examples
Understanding inequalities and equations is crucial in various real-life scenarios. For instance, when managing a budget, you might use inequalities to ensure your expenses stay below a certain limit. Similarly, in physics, equations are used to model the motion of objects, allowing us to predict their future positions. Mastering these concepts provides a strong foundation for problem-solving in many fields.
The solution to the inequality is -5"> x > − 5 and the solution to the equation is y = − 38 .
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