Consider the equations and inequalities below:
[tex]\begin{array}{l}<
\left(E_1\right):-2(x+3)=x+6 \\
\left(E_2\right): 2 x-\frac{3}{2}=\frac{7}{9} \\
\left(I_1\right): 5 x+3 \textless 10 \\
\left(I_2\right):-3 x+4 \textgreater -5 x+6
\end{array}[/tex]
1) Solve in [tex]$Q$[/tex] the equations [tex]$\left(E_1\right)$[/tex] and [tex]$\left(E_2\right)$[/tex]
2) Solve in [tex]$Q$[/tex] the inequalities ([tex]$I_1$[/tex] ) and ( [tex]$I_2$[/tex] )
3) Find for each inequality [tex]$\left(I_1\right)$[/tex] and [tex]$\left(I_2\right)$[/tex] three integer solutions
Answer (2)
The solutions are: for equation (E1), x = − 4 ; for equation (E2), x = 36 41 . For inequality (I1), the solution is x < 5 7 with integer solutions of 1, 0, and -1; for inequality (I2), the solution is 1"> x > 1 with integer solutions of 2, 3, and 4.
;
Solve the linear equation ( E 1 ) and find x = − 4 .
Solve the linear equation ( E 2 ) and find x = 36 41 .
Solve the linear inequality ( I 1 ) and find x < 5 7 . Three integer solutions are 1 , 0 , − 1 .
Solve the linear inequality ( I 2 ) and find 1"> x > 1 . Three integer solutions are 2 , 3 , 4 .
The solutions to the equations and inequalities are 1 (2, 3, 4)}"> E 1 : x = − 4 , E 2 : x = 36 41 , I 1 : x < 5 7 ( 1 , 0 , − 1 ) , I 2 : x > 1 ( 2 , 3 , 4 ) .
Explanation
Problem Analysis We are given two equations and two inequalities. Our task is to solve for x in each case and then find three integer solutions for each inequality.
Equations: ( E 1 ) : − 2 ( x + 3 ) = x + 6 ( E 2 ) : 2 x − 2 3 = 9 7
Inequalities: ( I 1 ) : 5 x + 3 < 10 -5x + 6"> ( I 2 ) : − 3 x + 4 > − 5 x + 6
Solving Equation (E1) Let's solve equation ( E 1 ) :
− 2 ( x + 3 ) = x + 6 − 2 x − 6 = x + 6 − 3 x = 12 x = − 4
So, the solution to equation ( E 1 ) is x = − 4 .
Solving Equation (E2) Now, let's solve equation ( E 2 ) :
2 x − 2 3 = 9 7 2 x = 9 7 + 2 3 2 x = 18 14 + 18 27 2 x = 18 41 x = 36 41
So, the solution to equation ( E 2 ) is x = 36 41 .
Solving Inequality (I1) Now, let's solve inequality ( I 1 ) :
5 x + 3 < 10 5 x < 7 x < 5 7 x < 1.4
Three integer solutions for ( I 1 ) are 1 , 0 , − 1 .
Solving Inequality (I2) Now, let's solve inequality ( I 2 ) :
-5x + 6"> − 3 x + 4 > − 5 x + 6 2"> 2 x > 2 1"> x > 1
Three integer solutions for ( I 2 ) are 2 , 3 , 4 .
Final Answer The solutions are: ( E 1 ) : x = − 4 ( E 2 ) : x = 36 41 ( I 1 ) : x < 5 7 , three integer solutions: 1 , 0 , − 1 1"> ( I 2 ) : x > 1 , three integer solutions: 2 , 3 , 4
Examples
Understanding how to solve equations and inequalities is crucial in many real-world scenarios. For example, imagine you are managing a budget and need to determine how much you can spend on different items while staying within your financial limits. Equations help you calculate exact amounts, while inequalities help you determine ranges of possible spending. Similarly, in science and engineering, these mathematical tools are used to model and solve problems related to resource allocation, optimization, and constraints.
