Question

# Solve absolute value inequality : abs(x + 3) <= 4

Piecewise-Defined Functions
Solve absolute value inequality : $$\displaystyle{\left|{{x}+{3}}\right|}\le{4}$$

2020-11-21
Step 1
the given absolute value inequality is:
$$\displaystyle{\left|{{x}+{3}}\right|}\le{4}$$
we have to solve the given inequality.
Step 2
the given absolute value inequality is $$\displaystyle{\left|{{x}+{3}}\right|}\le{4}$$
as we know that if $$\displaystyle{\left|{{x}}\right|}\le{a}$$ then $$\displaystyle{x}\in{\left[−{a},{a}\right]}$$ or $$\displaystyle−{a}\le{x}\le{a}$$
therefore,
$$\displaystyle{\left|{{x}+{3}}\right|}\le{4}$$
then,
$$\displaystyle-{4}\le{x}+{3}\le{4}$$
$$\displaystyle-{4}-{3}\le{x}+{3}-{3}\le{4}-{3}$$
$$\displaystyle-{7}\le{x}\le{1}$$
therefore we have $$\displaystyle{x}\in{\left[−{7},{1}\right]}$$
therefore the solution of the given absolute value inequality is $$\displaystyle{x}\in{\left[−{7},{1}\right]}$$