Question

What are the solutions to the inequality $\left( x-3 \right)\left( x+5 \right)\le 0$ ?

Answer 1

Hint: In the given question, we are supposed to find the solutions to the inequality $\left( x-3 \right)\left( x+5 \right)\le 0$ . We solve the given question by finding out the solutions of the linear expressions $\left( x-3 \right)\le 0$ and $\left( x+5 \right)\le 0$ to get the desired result.

Complete step by step answer:
We are given an inequality and are asked to find the solutions for the same. We will be solving the given question by solving the linear equations by isolating the variable x.
Algebraic expressions in mathematics are made up of variables, constants, and operators.
Every algebraic expression has variables, coefficients, constants, and terms.
Arithmetic operations like addition, subtraction, multiplication, and division can be performed on Algebraic expressions.
The like terms in algebra mean that the terms have the same variable and same power.
In Algebra, Only like terms can be added or subtracted.
The inequality in the question is given as follows,
\[\Rightarrow \left( x-3 \right)\left( x+5 \right) \le 0 \]
We find the value of x by solving the equations $(x-3)=0$ and $( x+5)=0$
Following the same, we get,
\[ \Rightarrow \left( x-3 \right)=0 \]
Adding the number 3 on both sides of the equation, we get,
\[ \Rightarrow \left( x-3 \right)+3=0+3\]
Simplifying the above equation, we get,
$\Rightarrow x-3+3=3$
$\therefore x=3$
Solving the equation $x+5=0$ , we get,
$\Rightarrow x+5=0$
Adding the number $-5$ on both sides of the equation, we get,
\[ \Rightarrow \left( x+5 \right)-5=0-5\]
Simplifying the above equation, we get,
$\Rightarrow x+5-5=-5$
$\therefore x=-5$
Now,
The given inequality $\left( x-3 \right)\left( x+5 \right)\le 0$ is satisfied only when the linear expressions $\left( x-3 \right)$ and $\left( x+5 \right)$ have opposite signs.
For any value of $x\le 3$ ,
The expressions $\left( x-3 \right)$ and $\left( x+5 \right)$ have opposite signs.
For any value of $x\ge -5$ ,
The expressions $\left( x-3 \right)$ and $\left( x+5 \right)$ have opposite signs.
From the above, the value of x can be written in the form of inequality as follows,
$\Rightarrow -5\le x\le 3$
The internal notation for the above inequality is given as follows,
$\Rightarrow x\in \left[ -5,3 \right]$
The graph of the inequality is given as follows,

From the graph,
We can say that the solutions to the inequality $\left( x-3 \right)\left( x+5 \right)\le 0$ are in the interval of $x\in \left[ -5,3 \right]$
$\therefore$ The solutions to the inequality $\left( x-3 \right)\left( x+5 \right)\le 0$ are in the internal of $x\in \left[ -5,3 \right]$

Note: The result obtained in the given question can be cross-checked by substituting the values of x in the inequality. If the result on substitution is lesser than or equal to zero, the inequality is satisfied and the result attained is correct.