Question

What is the range of $x$ in equation’s set of \[\left| {x + 2} \right| \leqslant 5\] ?

Answer 1

Hint: This question is based on inequalities, when we solve the inequality, we generally get a range of values not exact values like in the equation. Solving the given inequality is similar to solving equations in that we do the majority of the same things, but we must pay attention to the direction of inequality.

Complete step by step answer:
An inequality compares two values to determine whether one is less than, greater than, or simply not equal to the other.Direction of inequality does not change in these conditions: when from both sides we add or subtract a number. When both sides should be multiplied or divided by a positive value. We simplify one side.We have given,
\[\left| {x + 2} \right| \leqslant 5\]

We know that modulus of any number only includes its positive parts.
We remove modulus, it splits both in positive and negative.
$ - 5 \leqslant x + 2 \leqslant 5$
We subtract 2 from both sides
We are subtracting 2 from both sides. so, the direction of inequality will not change
$ - 5 - 2 \leqslant x \leqslant 5 - 2$
$\therefore - 7 \leqslant x \leqslant 3$

So, the range of x is from -7 to 3.

Note: The sign of an inequality is unaffected by adding or subtracting the same number from both sides of an inequation. Both sides of an inequation can be multiplied or divided by the same positive real number without changing the sign of the inequality, but when both sides of an inequation are multiplied or divided by a negative number, the sign of the inequality is reversed. Any term in an inequation can be transferred to the other side with its sign changed without changing the sign of the inequality.