Solving absolute value inequalities in interval notation :
Here we are going to see how to solve absolute value inequalities in interval notation.
If the given inequalities are in the following form, we may represent the the function inside the absolute value sign between the range -r and r and solve for x.
Given function |
Solution |
(i) |x - a| < r (ii) |x - a| > r (iii) |x - a| ≤ r (iv) |x - a| ≥ r |
(a - r, a + r) (∞, a - r) U (a + r, ∞) (∞, a - r] U [a + r , ∞) (∞, a - r] U [a + r, ∞) |
Here we may have some different cases.
Given function |
Solution |
|x - a| > -r (or) |x - a| ≥ - r |
Hence the solution is all real numbers.Since we have modulus sign in the left side, always we get positive value as answer. |
|x - a| < -r (or) |x - a| ≤ - r |
There is no solution. By applying any positive and negative values for x, we get only positive answer. Because we have modulus sign in the left side. |
Let us look into some example problems to understand the concept.
Question 1 :
Solve 1/|2x−1| < 6 and express the solution using the interval notation.
Solution :
Now we have to split the given inequality into two branches.
1/(2x-1) > -6 Multiply by 2x - 1 throughout the equation 1 > -6(2x - 1) 1 < -12x + 6 Subtract 6 on both sides 1 - 6 < -12x + 6 - 6 -5 < -12x Divide by -12 on both sides 5/12 > x |
1/(2x-1) < 6 Multiply by 2x - 1 throughout the equation 1 > 6(2x - 1) 1 < 12x - 6 Add 6 on both sides 1 + 6 < 12x - 6 + 6 7 < 12x Divide by 12 on both sides 7/12 < x |
In the last step of first part, we divide -12 throughout the equation. So < sign becomes > sign
Hence the solution in interval notation is (-∞, 5/12) U (7/12, ∞).
Question 2 :
Solve −3|x| + 5 ≤ −2 and graph the solution set in a number line.
Solution :
Let us split the given inequality into two parts
−3(x) + 5 ≤ −2 -3x + 5 ≤ −2 Subtract 5 on both sides -3x + 5 - 5 ≤ −2 - 5 -3x ≤ −7 Divide by -3 on both sides x ≥ 7/3 |
−3(-x) + 5 ≤ −2 3x + 5 ≤ −2 Subtract 5 on both sides 3x + 5 - 5 ≤ −2 - 5 3x ≤ −7 Divide by 3 on both sides x ≤ −7/3 |
The interval notation of the above solution is (-∞, -7/3] U [7/3, ∞).
After having gone through the stuff given above, we hope that the students would have understood "Solving absolute value inequalities in interval notation".
Apart from the stuff given above, if you want to know more about "Solving absolute value inequalities in interval notation", please click here
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits