Question

How do you solve the equation $4\left( 2-x \right)>-2x-3\left( 4x1 \right)$ ?

Answer 1

To solve the inequality we will first open the brackets in the equation by using the distributive property. Now we will separate the constants and the variables and then divide the inequality with the coefficient of ${{x}^{2}}$ hence we get the required condition on x.

Complete step-by-step solution:
Now let us consider the given equation $4\left( 2-x \right)>-2x-3\left( 4x1 \right)$
First let us consider the bracket $\left( 4x1 \right)$ Now we know that for any number x $x\times 1=x$ hence we can say that $\left( 4x1 \right)=\left( 4x \right)$ Hence we get the given equation as $4\left( 2-x \right)>-2x-3\left( 4x \right)$ .
Now we will open the brackets by using the distributive property $a\left( b+c \right)=ab+ac$ .
Hence we get the equation as,
$\Rightarrow 8-4x>-2x-12x$
Now taking 4x on RHS we get,
$\Rightarrow 8>4x-2x-12x$
Now since the powers of x are same we can add and subtract the terms, hence we get,
$\begin{aligned} & \Rightarrow 8>2x-12x \\\ & \Rightarrow 8>-10x \\\ & \Rightarrow 8>-10x \\\ \end{aligned}$
Now dividing the whole equation by – 10 we get, $-0.8< x$ .
Hence the solution of the given equation is $x\in \left( -0.8,\infty \right)$.

Note: Now note that solving equalities and inequalities are almost similar. Now in both cases we do algebraic operation like add terms subtract term multiply terms and divide terms. Now when solving equality we can do this without changing anything in the equation. But while solving inequality if we multiply or divide the equation by a negative term then the sign of inequality changes. For example suppose we have an equation $-x>9$ Now if we multiply the inequality by – 1 the sign changes and hence we get $x<-9$ . Hence always be careful while multiplying or dividing.