Question

How do you solve and graph $5x + 7 \leqslant 32$?

Answer 1

Hint: The given inequality equation is $5x + 7 \leqslant 32$
Eliminate fractions by multiplying all terms by the least common denominator of all fractions. Simplify by combining like terms on each side of the inequality.
Add or subtract quantities to obtain the unknown on one side and the numbers on the other.
We find the $x$ value and draw a graph.

Complete step-by-step solution:
The given inequality equation is $5x + 7 \leqslant 32$
We find the $x$ value
Let, $5x + 7 \leqslant 32$
First we have subtract $7$ on both sides, hence we get
$ \Rightarrow 5x + 7 - 7 \leqslant 32 - 7$
In LHS (Left Hand Side) subtract $7$ by $7$ and in RHS (Right Hand Side) subtract $32$ by $7$, hence we get
$ \Rightarrow 5x + 0 \leqslant 25$
The zero terms vanish
$ \Rightarrow 5x \leqslant 25$
Divide by $5$ on both sides, hence we get
\[ \Rightarrow \dfrac{{\not{5}}}{{\not{5}}}x \leqslant \dfrac{{25}}{5}\]
Divide $25$ by $5$, hence we get
$ \Rightarrow x \leqslant 5$
So you will see a straight line at $y = 1$from$ - \infty < x \leqslant 5$ and $y = 0$ from $5 < x < \infty $. This $1$ indicates that inequality is true and $0$ indicates that inequality is false. Just like how in computer language $1$ is true and $0$ is false.

Note: Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol$x = y$, $x$ is equal to $y$.
Where as in inequality, the two expressions are not necessarily equal which is indicated by the symbols: $ > , < , \leqslant $ or $ \geqslant $
$x > y$, $x$ is greater than $y$
$x \geqslant y$, $x$ is greater than or equal to $y$
$x < y$, $x$ is less than $y$
$x \leqslant y$, $x$ is less than or equal to $y$
An equation or an inequality that contains at least one variable is called an open sentence. When you substitute a number for the variable in an open sentence, the resulting statement is either true or false. If the statement is true, the number is a solution to the equation or inequality.