Why Linear Inequations Show Up Everywhere
Ever wondered why you can’t just buy a candy if you have less than $5? That little "less than" sign is an inequation, and solving it tells you whether you can afford the treat. In ICSE Class 10 Maths, linear inequations are the tool that turns everyday "if‑this‑then‑that" into a neat picture on a number line.
💡 In Simple Words: A linear inequation is like a balance scale that says one side must be bigger, smaller, or equal to the other. Solving it means finding all the numbers that keep the scale balanced the way the sign tells you.
Understanding Linear Inequations
A linear inequation looks just like a linear equation (which you solve for an exact value) except it uses symbols like , ≤ or ≥ instead of =. These symbols mean "less than", "greater than", "less than or equal to", and "greater than or equal to" respectively. For example, 3x − 2 asks for every x that makes the left side smaller than 7.
Think of the inequality as a fence. The fence tells you which side you’re allowed to walk on. Your job is to map out the whole safe region.
Steps to Solve a Linear Inequation
Just like a recipe, you follow a set order. Miss a step and the answer can slip.
1. Write the inequality
Copy the problem exactly as given. Any mistake here will throw everything off.
2. Move constants to one side
Just like solving an equation, add or subtract the same number on both sides to gather all the plain numbers together.
3. Isolate the variable
Divide or multiply to get the variable (usually x) alone. Remember, if you’re dealing with a fraction, you can multiply both sides by the denominator to clear it.
4. Flip the sign when needed
Here’s the tricky part: if you multiply or divide by a negative number, the direction of the inequality reverses. For example, if you have -2x > 6, dividing by -2 gives x . The sign flips from > to 5. Write the solution setUse interval notation (parentheses for ; brackets for ≤, ≥) or describe it in words. (-∞, 4) means “all numbers less than 4.”6. Test with a valuePick a number inside your answer and plug it back into the original inequation. If it works, you’re good.Graphing on a Number LineTurning the answer into a picture helps you see the range at a glance.Draw a horizontal line and mark a few key points.If the inequality is strict (), draw an open circle at the boundary point – the point itself isn’t allowed.If it’s non‑strict (≤ or ≥), draw a closed (filled) circle – the boundary point is included.Shade the line in the direction the inequality points. For x , shade leftwards; for x ≥ -2, shade rightwards.Example: Solve 2x + 5 ≥ -1.Step‑by‑step:Subtract 5: 2x ≥ -6.Divide by 2 (positive, so sign stays): x ≥ -3.Graph: draw a closed circle at -3 and shade everything to the right.Common Mistakes to AvoidForgetting to flip the sign when you multiply/divide by a negative number.Mixing up open and closed circles on the graph – an open circle means the point is not part of the solution.Leaving fractions unsimplified which can make the final interval look messy.Not checking a test value – a tiny arithmetic slip can give a wrong region.Quick Comparison TableActionEquation (=)Inequation (<, >, ≤, ≥)GoalFind one exact numberFind a range of numbersGraphSingle point on a lineShaded region with open/closed circleSign flip ruleNever neededFlip when multiplying/dividing by negative📝 Likely Exam QuestionsSolve and graph: 5x − 3 > 2x + 7.Solution: 5x − 3 > 2x + 7 ⇒ 3x > 10 ⇒ x > 10/3. Graph with an open circle at 3.33 and shade right.Find the solution set of −4 ≤ 3 − x.Solution: −4 ≤ 3 − x ⇒ −7 ≤ −x ⇒ 7 ≥ x ⇒ x ≤ 7. Interval: (‑∞, 7] with a closed circle at 7.If a student’s budget is less than $20, which values of y satisfy 2y + 5 Solution: 2y + 5 Shade left of 10 on the number line.Explain why the sign flips in the step from −3x ‑3.Answer: Dividing by –3 changes the direction of the inequality because multiplying both sides by a negative reverses the order.Graph the solution of x ≥ ‑2 and x Answer: Draw a closed circle at –2, an open circle at 4, and shade the region between them.