What are linear inequalities

If in an expression, two expressions or values are written equal, then it is called an equation. For instance, $$3x+4y=12$$.

On contrary to this, if an expression shows the relation between the two values with the sign ‘$$<$$’(less than), ‘$$>$$’ (greater than), ‘$$\leq$$’(less than or equal to), ‘$$\geq$$’(greater than or equal to), then it is termed as an inequality.

When the inequality is in the form of a linear function then it is termed as a linear inequality. It is similar to the linear equation; the only difference is that the sign of ‘$$=$$’ is changed with the sign of the inequalities. When solving the inequation, no matter what we do on both the sides of the sign, that is, we can multiply divide add, or subtract the inequality is true, but if we multiply or divide it by a negative number, the inequality sign changes to its opposite.

E2.5: Derive and solve linear inequalities

For solving linear inequalities, the same procedure is followed as for the linear equations. Some other rules that should be followed are stated above. Keep those rules in mind and proceed to manipulate the left-hand side and right-hand side of the inequality and obtain the desired value.

Worked examples of linear inequalities

Example 1: Solve the inequality $$17+5x<2$$ and present the solution in the number line.

Step 1: Subtract $$17$$ from both the sides to remove $$17$$ from the left side.

$$17+5x-17<2-17$$

$$5x&<-15$$

Step 2: Divide both the sides by $$5$$ to get the value of the unknown variable.

$$5x<-15$$

$$frac{5x}{x}<\frac{-15}{5}$$

Step 3: Simplify to get the solution.

$$x<-5$$.

Step 4: Draw the solution on the number line.

Example 2: For what values of $$x$$, the inequality $$-10<5(x+2)\leq25$$ is true? Represent the range of $$x$$ on the number line.

Step 1: First thing to notice that this is a combination of two inequalities, split the inequations and solve them independently.

The two independent inequations are $$-10<5(x+2)$$, $$5(x+2)\leq25$$

Step 2: Divide the first inequation by $$5$$ on both the sides to get the unknown value.

$$-10<5(x+2)$$

$$\frac{-10}{5} < \frac{5(x+2)}{5}$$

$$-2<x+2$$

Step 3: Subtract $$2$$ from both sides to get the value of the unknown.

$$-2-2<x+2-2$$

$$-4<x$$

Step 4: Divide the second inequation by $$5$$ on both the sides to get the unknown value.

$$\frac{5(x+2)}{5}\leq\frac{25}{5}$$

$$x+2\leq 5$$.

Step 5: Subtract $$2$$ from both the sides to get the value of the unknown.

$$x+2-2 \leq 5-2$$

$$x \leq 3$$

Step 6: Represent the solution on the number line.

$$-4<x \leq 3$$

Example 3: Stan has $$\$1000$$ in his saving bank account. He started withdrawing $$\$100$$ from his bank account every week for his weekly expenses. If he doesn’t make any deposit, how many weeks he can withdraw the amount, if he wants to maintain a balance of at least $$\$200$$?

Step 1: Assume the number of weeks for which he can withdraw the amount.

Let the number of weeks he can withdraw the amount be $$x$$.

Step 2: For the given condition, make an inequality.

According to question, the money left after withdrawing the amount should be greater than $$200$$. For the given condition, the equation becomes $$\text{starting balance}-\text{money withdrawn}\geq\$200$$

Step 3: Substitute the value of the starting balance as $$1000$$ and money withdrawn as $$100x$$.

$$1000-100x\geq 200$$

Step 4: Subtract $$1000$$ from both the sides.

$$1000-100x-1000 \geq 200-1000$$

$$-100x \geq -800$$.

Step 5: Divide both the sides by $$-100$$ and change the sign of the inequality.

$$\frac{-100x}{-100} \leq\frac{-800}{-100}$$

$$x \leq 8$$.

Thus, Stan can withdraw for 8 weeks to maintain a minimum balance of $$\$200$$ in his account.