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

Question

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

Julia Adams · Accepted Answer

#0<=x<=6#

#"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"#

Kennedy Adams · Answer

#0≤x≤6#, #x\in [0, 6]# 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]#

Julia Adams · Answer

undefined To solve the inequality $|3 - x| \leq 3$, follow these steps:

1. **Break the inequality into two cases**: 
   a) $3 - x \leq 3$ 
   b) $3 - x \geq -3$

2. **Solve each case separately**: 
   a) $3 - x \leq 3$:  
      $3 - x \leq 3$  
      $-x \leq 0$  
      $x \geq 3$

b) $3 - x \geq -3$:  
      $3 - x \geq -3$  
      $-x \geq -6$  
      $x \leq 6$

3. **Combine the solutions**:  
   $x \geq 3$ and $x \leq 6$  
   Therefore, the solution is $3 \leq x \leq 6$.

How do you solve the inequality #abs(3-x)&lt;=3#?

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