Free Absolute Value Inequalities Calculator

Solve absolute value inequalities instantly. Enter |x| < a, |x| > a, |x| <= a, or |x| >= a and get step-by-step interval solutions.

Solve absolute value inequalities with our free calculator. Enter any inequality involving |x| and get interval notation solutions with step-by-step explanations.

Key Takeaway

Use the free Absolute Value Inequalities Calculator to solve absolute value inequalities instantly. enter |x| < a, |x| > a, |x| <= a, or |x| >= a and get step-by-step interval solutions. Get instant results with step-by-step explanations.

How to Use the Absolute Value Inequalities Calculator

Enter the absolute value expression on the left side.
Select the inequality type: <, >, <=, or >=.
Enter the value or expression on the right side.
Review the solution in interval notation and number line representation.

The Formula

|expression| < a means -a < expression < a (AND, intersection). |expression| > a means expression < -a OR expression > a (OR, union). Same logic applies to <= and >=.

Variable Definitions

|x|: The absolute value of x
a: The boundary value (must be >= 0 for < and <=)
<: Less than — solution is inside the interval
>: Greater than — solution is outside the interval

Solving |x - 3| < 5

Find all values of x where the absolute value of (x - 3) is less than 5.

Step 1: Convert to compound inequality: -5 < x - 3 < 5.
Step 2: Add 3 to all parts: -5 + 3 < x < 5 + 3.
Step 3: Simplify: -2 < x < 8.
Step 4: In interval notation: (-2, 8).

Frequently Asked Questions

What is an absolute value inequalities calculator?

An absolute value inequalities calculator is an online algebra tool that solves inequalities containing absolute value expressions. It handles |A| < B, |A| ≤ B, |A| > B, and |A| ≥ B, returning the solution interval and often graphing it on a number line.

How do I use the absolute value inequality calculator?

Enter the inequality with absolute value bars, using ≤ or ≥ for inclusive bounds. For example, |x − 3| < 5. The calculator immediately shows the solution interval and graph for the variable.

What is the rule for |x| < B versus |x| > B?

For |x| < B, the calculator returns −B < x < B. For |x| > B, it returns x < −B or x > B. This direction flip is the most common student mistake the calculator avoids.

Can the calculator solve |x| ≤ 8?

Yes. The calculator returns the compound inequality −8 ≤ x ≤ 8, shaded on a number line with closed circles at both endpoints to show inclusivity.

Can the calculator solve |x| ≥ 3?

Yes. It returns x ≤ −3 or x ≥ 3. The number line graph shows arrows pointing outward from −3 and 3, not shaded between them.

How does the calculator handle inequalities with variables inside?

For |2x + 1| < 7, the calculator isolates the absolute value, splits to −7 < 2x + 1 < 7, subtracts 1, divides by 2, and returns −4 < x < 3.

Can this calculator show solution steps?

Yes. Step-by-step calculators display the compound inequality transformation: isolating the absolute value, splitting cases, solving each side, and writing the final interval notation.

Does the calculator graph the solution on a number line?

Many absolute value inequality calculators display an interactive number line with the solution region shaded, including open or closed circles for strict or inclusive inequalities.

Can the calculator solve compound absolute value inequalities?

Yes. Double inequalities like −3 ≤ |x − 2| ≤ 5 are solved by finding the intersection of two separate inequalities. The calculator handles both bounds automatically.

What if the right side of the inequality is negative?

|x| < −4 has no solution because absolute value is always non-negative and cannot be less than a negative number. The calculator returns "no solution" with explanation.

Can the calculator solve absolute value inequalities with fractions?

Yes. Enter |x/2 + 1/3| ≤ 5/3 and the calculator works through the arithmetic with fractions, returning an exact fractional or decimal interval.

How does the calculator handle compound inequalities with AND?

For |x − 2| < 4 AND |x + 1| > 1, the calculator finds the solution intervals of both inequalities and returns their intersection.

Can the calculator handle compound inequalities with OR?

Yes. For |x − 1| > 3 OR |x + 2| < 2, the calculator returns the union of both solution sets, showing all values that satisfy either condition.

Does the calculator return interval notation?

Yes. A typical result like (−4, 3) is shown alongside set-builder notation {x | −4 < x < 3} for complete mathematical clarity.

Can I use this calculator for algebra 2 homework help?

Yes. Absolute value inequalities are core algebra 2 content. The calculator lets students check manual solutions and learn the case-splitting method through visible steps.

How does absolute value inequality relate to word problems?

"Within 5 miles of home" translates to |d| ≤ 5. The calculator converts these distance phrases directly into algebraic inequality solutions.

Can the calculator solve absolute value inequalities from tolerance problems?

Yes. If a manufactured part must be within |d − 10| ≤ 0.05 mm, the calculator returns 9.95 ≤ d ≤ 10.05, which engineers use as quality control bounds.

How does the calculator help with temperature range problems?

If a vaccine must be stored within 2°C of 5°C, |T − 5| ≤ 2 is entered and the calculator returns 3 ≤ T ≤ 7, giving the safe storage range.

Can I use this calculator for error margin problems in science?

Yes. If a measurement must be within |x − 10| < 0.5 of a target, the calculator returns 9.5 < x < 10.5, showing acceptable measurement tolerance.

How does the calculator help with safety margin problems in engineering?

If stress must stay within |σ − σ_allowable| ≤ 10% of allowable, the calculator gives the safe operational stress range the engineer uses in design.

Can I use this calculator for business budget problems?

Yes. If spending must be within |S − 5000| ≤ 500 of a $5,000 budget, the calculator returns 4,500 ≤ S ≤ 5,500, giving the acceptable spending window.

How does the calculator help with quality control limits?

If product weight must be within |w − 200| ≤ 3 grams, the calculator returns 197 ≤ w ≤ 203, which production teams use for acceptable product ranges.

Can I use this calculator for grading tolerance problems?

Yes. If a test score must be within 8 points of 85, |x − 85| ≤ 8 returns 77 ≤ x ≤ 93, showing the acceptable score range for a grade of B.

How does the calculator help with delivery time estimates?

If pizza must arrive within |t − 30| ≤ 5 minutes, the calculator returns 25 ≤ t ≤ 35, defining the guaranteed delivery window.

Can I use this calculator for sports performance problems?

Yes. If a runner's time must be within 2 minutes of 18, |t − 18| ≤ 2 returns 16 ≤ t ≤ 20, giving the acceptable time window for team selection.

How does the calculator help with financial deviation problems?

If actual sales must be within |S − 100000| ≤ 15000 of a target, the calculator returns 85,000 ≤ S ≤ 115,000, defining acceptable revenue outcomes.

Can I use this calculator for polling margin of error problems?

Yes. If a poll result is 52% with margin of error ±3%, |p − 0.52| ≤ 0.03 returns 49% ≤ p ≤ 55%, giving the confidence interval.

How does the calculator help with temperature control problems?

If an oven must stay within |T − 350| ≤ 10°F, the calculator returns 340 ≤ T ≤ 360, defining the safe cooking temperature range.

Can I use this calculator for medication dosage problems?

Yes. If a patient must stay within |dose − 500| ≤ 50 mg of target, the calculator returns 450 ≤ dose ≤ 550, giving the safe medication range.

How does the calculator help with blood sugar management problems?

If blood glucose must be within |G − 120| ≤ 30 mg/dL, the calculator returns 90 ≤ G ≤ 150, showing the acceptable blood sugar window.

Can I use this calculator for parking meter problems?

Yes. If parking time must be within |t − 120| ≤ 15 minutes, the calculator returns 105 ≤ t ≤ 135, defining legal parking duration before meter expiry.

How does the calculator help with oil change interval problems?

If oil changes are needed every 5000 miles with |miles − 5000| ≤ 500 tolerance, the calculator returns 4,500 ≤ miles ≤ 5,500, defining acceptable change windows.

Can I use this calculator for battery charge problems?

Yes. If a battery is considered low at |charge − 20| ≥ 10%, solving the boundary |charge − 20| = 10 gives the 10% threshold warning level.

How does the calculator help with credit score problems?

If a loan approval requires |score − 700| ≤ 100, the calculator returns 600 ≤ score ≤ 800, giving the credit score range for favorable lending terms.

Can I use this calculator for insurance deductible problems?

Yes. If a deductible tier requires |claim − 1000| ≤ 500, the calculator returns 500 ≤ claim ≤ 1,500, defining the payout range for that deductible level.

How does the calculator help with utility bill problems?

If a budget billing plan allows |bill − 150| ≤ 30, the calculator returns 120 ≤ bill ≤ 180, defining the monthly charge range for budget billing.

Can I use this calculator for shipping weight problems?

Yes. If a package surcharge applies when |weight − 50| ≥ 5 lbs, the boundary |weight − 50| = 5 gives 45 lbs and 55 lbs as surcharge thresholds.

How does the calculator help with airline baggage allowance problems?

If carry-on must be |weight − 40| ≤ 5 lbs, the calculator returns 35 ≤ weight ≤ 45, giving the acceptable carry-on weight range.

Can I use this calculator for hotel occupancy problems?

If overbooking is triggered when |occupancy − 95| ≥ 3%, the boundary equation gives the 92% and 98% thresholds for revenue management decisions.

How does the calculator help with concert ticket pricing problems?

If resale prices within |price − 100| ≤ 20 are fair, the calculator returns 80 ≤ price ≤ 120, defining the acceptable ticket resale range.

Can I use this calculator for gambling odds problems?

If a game is considered fair when |house_edge − 0| ≤ 0.02, the boundary gives acceptable house edge ranges for regulatory compliance.

How does the calculator help with sports betting problems?

If a spread of |margin − 7| ≤ 3 is a push, solving the boundary reveals the exact score combinations resulting in a tie on the spread.

Can I use this calculator for loan approval problems?

If DTI ratio must satisfy |DTI − 0.36| ≤ 0.06, the calculator returns 0.30 ≤ DTI ≤ 0.42, giving the acceptable debt-to-income range for mortgage approval.

How does the calculator help with investment risk tolerance?

If portfolio allocation to stocks must be |stocks − 0.6| ≤ 0.2 based on risk profile, the calculator returns 0.4 to 0.8, giving the acceptable stock allocation range.

Can I use this calculator for retirement withdrawal problems?

If annual withdrawal must be |withdrawal − 40000| ≤ 5000, the calculator returns $35,000 to $45,000, giving the safe annual spending band.

How does the calculator help with inflation target problems?

If the Fed targets |inflation − 0.02| ≤ 0.01, the calculator returns 0.01 to 0.03, giving the 1% to 3% acceptable inflation band.

Can I use this calculator for pH range problems in chemistry?

If a solution must have |pH − 7| ≤ 0.5, the calculator returns 6.5 ≤ pH ≤ 7.5, giving the acceptable acidity range for neutral solutions.

How does the calculator help with solution molarity problems?

If a lab solution must be |M − 0.1| ≤ 0.005 M, the calculator returns 0.095 ≤ M ≤ 0.105, giving the acceptable concentration range for titration.

Can I use this calculator for buffer capacity problems?

Yes. If a buffer maintains |pH − target| ≤ 0.1, the calculator gives the buffer capacity and the acceptable pH range for the experiment.

How does the calculator help with enzyme activity problems?

If enzyme rate must be |v − v_optimal| ≤ 10% of optimal, the calculator returns the acceptable temperature and pH range for optimal activity.

How does the calculator help with cell culture incubation problems?

If CO2 incubator must be |CO2 − 5| ≤ 0.5%, the calculator returns 4.5% to 5.5%, defining acceptable gas concentration for cell viability.

Can I use this calculator for PCR temperature control problems?

Yes. If annealing temperature must be |T − 55| ≤ 2°C, the calculator returns 53°C to 57°C, giving the acceptable temperature range for primer binding.

How does the calculator help with antibiotic resistance testing problems?

If zone of inhibition must be |zone − 20| ≥ 0 mm for resistance, the boundary |zone − 20| = 0 gives the minimum clear zone required for sensitivity classification.

Can I use this calculator for vaccine efficacy confidence intervals?

Yes. If efficacy is 90% with |efficacy − 0.90| ≤ 0.05, the calculator returns 85% to 95%, giving the acceptable efficacy confidence interval.

How does the calculator help with drug half-life problems?

If drug concentration must be |C − C_effective| ≤ 20% for therapeutic range, the calculator gives the acceptable peak and trough levels for dosing.

Can I use this calculator for blood pressure management problems?

Yes. If target BP is 120/80 with |systolic − 120| ≤ 10, the calculator returns 110 ≤ systolic ≤ 130, giving the acceptable blood pressure range.

How does the calculator help with cholesterol management problems?

If LDL target is 100 with |LDL − 100| ≤ 30, the calculator returns 70 ≤ LDL ≤ 130, giving the acceptable cholesterol range for heart health.

Can I use this calculator for thyroid hormone management problems?

Yes. If TSH target is 2.5 with |TSH − 2.5| ≤ 1.5, the calculator returns 1.0 ≤ TSH ≤ 4.0, giving the acceptable thyroid function range.

How does the calculator help with diabetes A1C management problems?

If A1C target is 6.5% with |A1C − 6.5| ≤ 1.5%, the calculator returns 5.0% to 8.0%, giving the glycemic control range for patients.

Can I use this calculator for fertility tracking problems?

Yes. If basal body temperature must be |BBT − 97.5| ≥ 0.4 to confirm ovulation, the calculator gives the temperature rise threshold for fertility awareness.

How does the calculator help with pregnancy weight gain problems?

If recommended weight gain is |weight − 25| ≤ 15 lbs for BMI 20–25, the calculator returns 10 to 40 lbs as the acceptable pregnancy weight gain range.

Can I use this calculator for newborn Apgar score problems?

If Apgar score must be |score − 7| ≤ 2 at 1 minute, the calculator returns 5 to 9, giving the acceptable newborn health score range.

How does the calculator help with child growth chart problems?

If height must be |height − 50th_percentile| ≤ 10 percentile points for normal growth, the calculator defines the acceptable growth trajectory range.

Can I use this calculator for immunization schedule problems?

Yes. If a vaccine must be given within |age − 12| ≤ 2 months, the calculator returns 10 to 14 months, defining the acceptable vaccination window.

How does the calculator help with hearing test problems?

If hearing threshold must be |dB − 20| ≤ 10, the calculator returns 10 to 30 dB, giving the acceptable hearing level range across frequencies.

Can I use this calculator for vision acuity problems?

Yes. If visual acuity must be |acuity − 20/20| ≤ 2 lines, the calculator defines the acceptable corrected vision range for driving eligibility.

How does the calculator help with lung function test problems?

If FEV1 must be |FEV1 − predicted| ≤ 15% for normal, the calculator returns the acceptable lung function range for spirometry interpretation.

Can I use this calculator for bone density problems?

If T-score must be |T − 0| ≤ 1 for normal bone density, the calculator returns −1 ≤ T ≤ 1, defining the normal range on the DEXA scan scale.

How does the calculator help with liver enzyme problems?

If ALT must be |ALT − 30| ≤ 20 U/L for normal, the calculator returns 10 to 50 U/L, giving the acceptable liver enzyme range.

Can I use this calculator for kidney function problems?

Yes. If GFR must be |GFR − 90| ≤ 15 for normal, the calculator defines the acceptable kidney function range in mL/min per 1.73m².

How does the calculator help with vitamin D level problems?

If 25(OH)D must be |level − 30| ≤ 10 ng/mL for sufficiency, the calculator returns 20 to 40 ng/mL, giving the acceptable vitamin D range.

Can I use this calculator for iron deficiency screening problems?

Yes. If ferritin must be |ferritin − 50| ≥ 0 ng/mL for women, the boundary gives the threshold level below which iron deficiency is diagnosed.

How does the calculator help with blood pressure medication problems?

If home BP readings must be |BP − 130| ≤ 10 for well-controlled hypertension, the calculator returns 120 to 140, defining the acceptable home reading range.

Can I use this calculator for insulin dosing problems?

Yes. If correction factor requires |blood_sugar − 120| ÷ 30 = units, the boundary gives the insulin dose range for sliding scale regimens.

How does the calculator help with anticoagulation therapy monitoring?

If INR must be |INR − 2.5| ≤ 0.5 for atrial fibrillation, the calculator returns 2.0 to 3.0, giving the therapeutic INR range.

Can I use this calculator for cancer tumor marker problems?

If PSA must be |PSA − 4| ≤ 1 for normal, the calculator returns 3 to 5 ng/mL, giving the acceptable prostate-specific antigen range.

How does the calculator help with HIV viral load problems?

If undetectable is |viral_load − 0| ≤ 20 copies/mL, the boundary gives the definition of viral suppression on modern ART therapy.

Can I use this calculator for HbA1c target problems?

If diabetes control target is |A1C − 7.0| ≤ 1.5%, the calculator returns 5.5% to 8.5%, giving the acceptable long-term glucose control range.

How does the calculator help with thyroid medication dosing problems?

If TSH must be |TSH − 2.0| ≤ 1.0 for hypothyroidism treatment, the calculator returns 1.0 to 3.0, defining the acceptable thyroid replacement range.

Can I use this calculator for glaucoma eye pressure problems?

If IOP must be |IOP − 21| ≤ 5 mmHg for normal, the calculator gives the acceptable intraocular pressure range to prevent optic nerve damage.

How does the calculator help with cholesterol ratio problems?

If total/HDL ratio must be |ratio − 3.5| ≤ 1.0, the calculator returns 2.5 to 4.5, giving the acceptable cardiovascular risk ratio.

Can I use this calculator for liver enzyme ALT problems?

If ALT must be |ALT − 25| ≤ 15 U/L, the calculator returns 10 to 40 U/L, giving the acceptable liver enzyme range for routine screening.

How does the calculator help with creatinine clearance problems?

If CrCl must be |CrCl − 120| ≤ 25 mL/min for young adults, the calculator defines the acceptable kidney filtration rate range by age.

Can I use this calculator for fasting glucose target problems?

If fasting glucose must be |glucose − 100| ≤ 20 mg/dL for normal, the calculator returns 80 to 120 mg/dL, giving the acceptable fasting blood sugar range.

How does the calculator help with postprandial glucose problems?

If 2-hour post-meal glucose must be |glucose − 140| ≤ 40 mg/dL, the calculator returns 100 to 180, defining acceptable post-meal blood sugar targets.

Can I use this calculator for ketone level problems?

If urinary ketones must be |ketones − 0| ≤ trace for normal, the boundary gives the acceptable ketone threshold for diabetic ketoacidosis screening.

How does the calculator help with urinary protein problems?

If urine protein must be |protein − 0| ≤ 30 mg/day for normal, the boundary gives the acceptable proteinuria threshold for kidney health screening.

Can I use this calculator for red blood cell count problems?

If RBC must be |RBC − 5.0| ≤ 1.0 million/µL for women, the calculator gives the acceptable red blood cell count range for anemia screening.

How does the calculator help with white blood cell count problems?

If WBC must be |WBC − 7.5| ≤ 3.0 thousand/µL for normal, the calculator returns 4.5 to 10.5, giving the acceptable immune function range.

Can I use this calculator for platelet count problems?

If platelets must be |platelets − 250| ± 50 thousand/µL, the calculator gives the acceptable platelet range for blood clotting function.

How does the calculator help with blood sodium problems?

If sodium must be |Na − 140| ≤ 5 mEq/L for normal, the calculator returns 135 to 145, giving the acceptable electrolyte range.

Can I use this calculator for blood potassium problems?

Yes. If potassium must be |K − 4.0| ± 0.5 mEq/L for normal, the calculator gives the acceptable potassium range for heart rhythm stability.

How does the calculator help with blood calcium problems?

If calcium must be |Ca − 9.5| ± 0.5 mg/dL for normal, the calculator returns the acceptable calcium range for bone and nerve health.

Can I use this calculator for thyroid TSH problems?

If TSH must be |TSH − 2.5| ≤ 2.0 for normal thyroid function, the calculator gives 0.5 to 4.5, defining the acceptable TSH testing range.

How does the calculator help with vitamin B12 problems?

If B12 must be |B12 − 400| ≥ 0 pg/mL for normal, the boundary gives the threshold below which supplementation is recommended.

Can I use this calculator for iron ferritin problems?

Yes. If ferritin must be |ferritin − 50| ≥ 0 ng/mL for women, the boundary defines iron sufficiency and deficiency thresholds.

How does the calculator help with folate level problems?

If folate must be |folate − 10| ≥ 0 ng/mL for normal, the boundary gives the threshold below which supplementation is recommended for women of childbearing age.

Can I use this calculator for vitamin D screening problems?

If 25(OH)D must be |vitamin D − 30| ≥ 0 ng/mL for sufficiency, the boundary gives the minimum level for bone health and immune function.

How does the calculator help with cortisol level problems?

If morning cortisol must be |cortisol − 15| ± 5 µg/dL for normal, the calculator gives the acceptable stress hormone range for adrenal function testing.

Why do students and teachers prefer an absolute value inequalities calculator with steps?

The step-by-step absolute value inequalities calculator transforms confusing sign-flip rules into transparent case-by-case algebra. Teachers recommend it for building intuition, students use it for homework verification, and both appreciate the visual number-line graph that makes solution intervals click instantly. It is one of the most searched free algebra calculators on the web for algebra 1, algebra 2, and college algebra courses.

Absolute Value Inequalities Calculator – Guide & Formulas

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