Question

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

Piecewise-Defined Functions
ANSWERED
asked 2020-11-20
Solve absolute value inequality : \(\displaystyle{\left|{{x}+{3}}\right|}\le{4}\)

Answers (1)

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 \(\displaystyle{x}\in{\left[−{7},{1}\right]}\)
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