Question

Find the integral value of x for which $\left| {2x + 5} \right| \leqslant 9$

Answer 1

Hint- In order to solve the given question expand the inequality given in form of modulus function by the help of property of modulus function and the formula for the range of modulus function.

Complete step-by-step answer:

As we know that for any general modulus function given in the form of inequality as $\left| {f\left( x \right)} \right| \leqslant a$
Then the range of the function without modulus can be expressed as $ - a \leqslant f\left( x \right) \leqslant a$
Given function is $f\left( x \right) = 2x + 5$ and $a = 9$
Using the property given we get
$
  \because - a \leqslant f\left( x \right) \leqslant a \\
   \Rightarrow - 9 \leqslant 2x + 5 \leqslant 9 \\
 $
Now let us solve the inequality in order to find the range of x
Lets us first subtract 5 from the inequality
$
   \Rightarrow - 9 - 5 \leqslant 2x + 5 - 5 \leqslant 9 - 5 \\
   \Rightarrow - 14 \leqslant 2x \leqslant 4 \\
 $
Now let us divide the whole inequality by 2
\[
   \Rightarrow \dfrac{{ - 14}}{2} \leqslant \dfrac{{2x}}{2} \leqslant \dfrac{4}{2} \\
   \Rightarrow - 7 \leqslant x \leqslant 2 \\
 \]
Since the value of x lies between -7 and 2 so the integral value of x is all the integers from -7 to 2.
That is $ - 7, - 6, - 5, - 4, - 3, - 2, - 1,0,1,2$
Hence, the integral values of x satisfying the inequality is $\left[ { - 7,2} \right]$

Note- In order to solve such types of problems students must remember formulas for modulus function. This problem can also be solved by the graphical method. Students must remember the steps to solve inequality. If we multiply or divide an inequality by a negative number, the sign of inequality reverses.