The MathBlog Absolute Value Inequalities Calculator helps you quickly solve absolute value inequalities, providing detailed steps to enhance your understanding and learning. Just put in your coefficients below and get detailed steps on how to solve modulus inequalities.

## Absolute Value Inequality Calculator

a∙|b∙x + c| + d < e

### How to use the MathBlog absolute value inequalities calculator

Our tool helps you solve inequalities of the form $$a∣bx+c∣+d$$. Here’s how it works:

1. Select the inequality sign: >, ≥, ≤, <. For = we have the absolute value equations calculator.
2. Input the coefficients a, b, c, d, and e for your inequality.
3. Click Calculate. The calculator will display your entered inequality below the input fields for verification.

## Solving Modulus Inequalities

To solve absolute value inequalities of the forms:

• $$a∣b x + c∣+d > e$$
• $$a∣b x + c∣+d ≥ e$$
• $$a∣b x + c∣+d < e$$
• $$a∣b x + c∣+d ≤ e$$

1. Simplify the Inequality: Subtract $$d$$ from both sides and divide by $$a$$. In case of > it looks like this
• $$|b x + c | > \frac {e-d}{a}$$
• NOTE: If $$a<0$$, remember to flip the inequality sign
2. Check the Right-Hand Side:
• For > (or ≥) signs:
• If the right-hand side is negative, all real numbers satisfy this inequality. That’s beacause the absolute value is always non-negativ.
• If the right-hand side is positive, there is at least one solution.
• For < (or ≤) signs
• If the right-hand side is negative, there are no solutions as an absolute value is always positive
• If the right-hand side is positive, there is at least one solution.
• If the right hand side is 0 and the sign is <, there’s no solution as the absolute value should be smaller than a negative number
• if the right-hand side is 0 and the is ≤, it becomes an equation where $$|b x + c | =0, x = – \frac {c}{b}$$
3. Calculate: Considering the points at #2, start calculating and figure out the values for $$x$$.
4. Express the Solution:
• Finally, rewrite the solution in interval notation for clarity.