How do you solve the inequality #abs(3-x)<=3#?

2 Answers
Jim G.
Mar 29, 2017

#0<=x<=6#

Explanation:

#"Given the inequality " |x|<=a#

#"the solution is always of the form " -a<=x<=a#

#rArr-3<=color(red)(3-x)<=3#

Isolate x in the centre interval while obtaining numeric values in the 2 end intervals.

subtract 3 from ALL intervals.

#-3-3<=cancel(3)cancel(-3)-x<=3-3#

#rArr-6<=-x<=0#

multiply by - 1 to obtain x

#color(blue)"Note"# when multiplying/dividing an inequality by a #color(blue)"negative"# value we must #color(red)" reverse the inequality symbol"#

#rArr6>=x>=0larrcolor(red)" reverse symbol"#

#rArrx<=6color(red)" and " x>=0#

#rArr0<=x<=6" is the solution"#

#x in [0,6]larrcolor(red)" in interval notation"#

NickTheTurtle
Mar 29, 2017

#0≤x≤6#, #x\in [0, 6]#

Explanation:

We can rewrite the equation as #-3≤3-x≤3#. Subtract both sides by #3# to get #-6≤-x≤0#. Now, we multiply both sides by #-1# and flip the inequality signs: #6≥x≥0#. This can be rewritten to #0≤x≤6#.

Other notations for this answer: #x\in [0, 6]#

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