Solving an absolute value inequality is similar to solving an absolute value equation,
Now, let's formalize these observations into a more mathematical statement: Absolute Value Inequalities:
 Solve for x: | x - 3 | < 4 [Working with "less than or equal to"]
 Solve for x: | x - 20 | > 5 [Working with "greater than"]
 Solve for x: | 3 + x | - 4 < 0 [Isolate absolute value.]
Separate a compound inequality into two separate problems.
Now, find where the solutions overlap! Solution: -8 < x < -6 as well as 4 < x < 6
 Solve for x: | x + 4 | > -3 [All values work.]
 Solve for x: | x + 1 | < -6 [No values work.]
 [word problem]It is reported that the average yearly salary for computer programmers in the United States is $51,423 per year, but can vary depending upon location. The actual salary could differ from the average by as much as $15,559 per year. a) Write an absolute value inequality to describe this situation. b) Solve the inequality to find the range of the starting salaries. Solution: a) | x - 51423 | < 15559
Answer: $35,864 < x < $66,982
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FAQs
Absolute Value Inequalities- MathBitsNotebook(A2)? ›
Absolute Value Inequalities- MathBitsNotebook(A2) | x - 0 | < 2 represents the distance between x and 0 that is less than 2. The solution will be all of the values on the number line between -2 and +2. | x - 0 | > 2 represents the distance between x and 0 that is greater than 2.
What is the absolute value of a number in MathBitsNotebook? ›Absolute Value - MathBitsNotebook(A1) The absolute value of a number is the distance between the number and zero on the real number line. Distances are measured as positive units (or zero units). Consequently, absolute value is never negative.
How to solve absolute value inequalities notes? ›To solve an absolute value inequality involving “less than,” such as |X|≤p, replace it with the compound inequality −p≤X≤p and then solve as usual. To solve an absolute value inequality involving “greater than,” such as |X|≥p, replace it with the compound inequality X≤−p or X≥p and then solve as usual.
What is an absolute value inequality in algebra 2? ›Absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols. In simple words, we can say that an absolute value inequality is an inequality with an absolute value symbol in it.
Can an absolute value inequality equal zero? ›If the absolute value is greater than or equal to zero, the solution is all real numbers. d. If the absolute value is greater than zero, the solution is all real numbers except for the value which makes it equal to zero.
How do I solve for absolute value? ›- Isolate the absolute value on one side of the equation.
- Is the number on the other side of the equation negative? ...
- Write two equations without absolute values. ...
- Solve the two equations.
The absolute value is the distance between a number and zero. The distance between −71 and 0 is 71 .
What is the first step to solve absolute value inequalities? ›| Step 1: Isolate the absolute value | |x + 4| - 6 < 9 |x + 4| < 15 |
|---|---|
| Step 2: Is the number on the other side negative? | No, it's a positive number, 15. We'll move on to step 3. |
| Step 3: Set up a compound inequality | The inequality sign in our problem is a less than sign, so we will set up a 3-part inequality: -15 < x + 4 < 15 |
An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative. You can write an absolute value inequality as a compound inequality. This holds true for all absolute value inequalities. You can replace > above with ≥ and < with ≤.
Can you solve absolute value inequalities by graphing? ›Solve two variable absolute value inequalities by graphing them. Graph the corresponding absolute value equation. Use a solid line if the inequality includes 'or equal to' or a broken line if it does not. Shade in the appropriate area above or below the line.
What is a real life example of an absolute value inequality? ›
These inequalities involve the absolute value of an expression containing a variable. Through real-world examples, such as determining the acceptable weight range for chocolate bars or understanding the temperature variations on Mars, one can grasp the practical applications of these inequalities.
What does absolute value mean in algebra 2? ›The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
How to prove inequalities with absolute values? ›Absolute Value Inequalities are usually proved by the absolute value being greater than or equal to a certain value. The square of the value is equal to the square of its absolute value.
How to solve inequality? ›When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed. So the solution is x > −1.
What is the third step in solving the absolute value inequalities and equations? ›Isolate the absolute value expression. Write the equivalent compound inequality. Solve the compound inequality. Write the solution using interval notation.
How to simplify absolute value? ›So, |x| means the absolute value of x, or modulus x. An absolute value expression is an expression that uses the absolute value symbol. To simplify an absolute value expression, write the expression in its simplest form. To do this, the absolute value symbol will be temporarily treated as a set of parentheses.
What is the absolute value of a number in math? ›Definitions: The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.
What will be the absolute value of a number? ›The absolute value of a number represents its distance from zero on a number line, always resulting in a positive value. This concept is essential in mathematics, as it helps to simplify calculations and understand the magnitude of numbers, regardless of their positive or negative sign.
What is the absolute value of |- 82? ›The absolute value is the distance between a number and zero. The distance between −82 and 0 is 82 .