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Linear Inequality Word Problems

In plain terms an inequality is opposite of an equation. Means an inequality states that two statements are not same or not only same. In real life we face of more inequalities than equations. Suppose the passing score in an examination is $40$ in a scale of $100$, it is not that you have to score exactly. You are through if you secure any score greater than or equal to $40$. If the speed limit in a highway is $60$ mph, the cops will not book you if you drive at $50$ mph. Of course, there are many situations where you need to operate between two limits, one lower and the other higher. The topic ‘Inequalities’ is the mathematical representation of these facts. In real life situations we come across many constraints which can be mathematically expressed. By solving them we find the way to get over the practical constraints.

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Definition

As already mentioned an inequality means that two statements are not same. Mathematically it is defined as two expressions are not equal. To accommodate a wider option, it may be refined as that the two expressions are not only equal but can also differ, For example if we say '$x$' is the score of a student and if $40$ is the passing level, then the inequality is '$x$ is greater than or equal to' $40$. On the other hand if the condition for passing is more than $40$ points, we need to say '$x$ is greater than' $40$. The difference is, the term 'or equal to' is missing in the second case. Such types of inequalities are called as strict inequalities.

Now let us define the mathematical symbols to be used in inequalities like we use '$=$' in case of equations. The symbol '$\geq$' is used for conveying the expression on the left side is 'greater than or equal to' the expression on the right side. In case the inequality is strict, the symbol '$>$' is used. It may be noted the bar is missing in the latter case. In the opposite situations the symbol direction is swapped. That is '$<$' means 'less than'. It is seen that the vertex of the arrow head is always towards the expression of lower value and this fact is used as a memory aid.

So, an inequality is a mathematical statement just like an equation and can be solved for the unknown variable, except for the conclusion of the results. In all cases the solutions of inequalities are not unique and are spread over an interval or intervals.  A graphical representation of the solutions greatly helps to understand clear and better.

Formula

The formulas that are used for solving an equation are also applicable in solving except for certain operations. You can add or subtract any integer on both sides but when you multiply or divide by a non- zero negative integer the inequality symbol has to be reversed. Similarly, you need to reverse the inequality symbol if you flip the expressions on both sides. For example, $5 < 6$ but when multiplying or dividing by $-1$ on both sides, $-5$ has to be $> -6$. Similarly, $4 > 2$ but when both sides flipped we see that $\frac{1}{4}$ $<$ $\frac{1}{2}$

In graphing the solutions we use open circles in case of strict inequalities and use closed circles otherwise. In the following examples this practice can be understood easily.

Examples

Example 1:

A team of husband and wife take part in a quiz. The wife scored $2$ points more than the husband. If both of them together scored less than 10 points, what could be the possible scores of the husband?

Solution:

Let '$x$' be the points scored by the husband and '$y$' be the same for the wife. As per the problem statement,

$y$ = $x + 2$ and $x + y < 10$

Plugging in $y$ = $x + 2$ in the inequality, $x + x + 2 < 10$ or $2x + 2 < 10$ or, $2x < 8$ or $x < 4$

Since the points are only positive integers, $x$ can be $0, 1, 2$ or $3$.

So, the possible scores of the husband are $0, 1, 2$ or $3$.
Example 2:

A ball is thrown up from ground and its height at any time '$t$' is given by the function $h(t)$ = $-1 6t^2 + 40t$. Find the time interval in which the height of the ball is greater than $16$ feet.

Solution:

As per the problem statement we need to find the time when $h(t) > 16$

In other words, $-16t^2 + 40t > 16$

Or, $-16t^2 + 40t -16 > 0$

Or, $2t^2 - 5t + 2 < 0$      (dividing throughout by $-8$ and reversing the inequality symbol)

Or, $2t(t - 2) -1(t - 2) > 0$

Or, $(2t - 1)(t - 2) > 0$

As per the negative product property, either $(2t - 1) > 0$ and $(t - 2) < 0$ or $(2t - 1) < 0$ and $(t - 2) > 0.$ But the second set of condition is Impossible because '$t$' cannot be less than $0.5$ and at the same time cannot be $> 2$. Hence, as per the first set of conditions $t > 0.5$ and $t < 2$. Therefore, the ball is at a height greater than $16$ feet between $0.5$ and $2$ seconds.  The solution is graphed on a number line as shown below.

The shaded part of the number line gives the time interval of the height of the ball greater than $16$ feet.
Example 3:

Robert and Ben run taxis. Robert charges $\$1.75$fixed and$ \$0.65$ per mile whereas Ben's terms are $\$ 2.50$fixed and$ \$0.50$ per mile. From how many miles of journey hiring Ben's taxi will be cheaper?

Solution:

Let '$x$' be the number of miles of journey and $y_1$ be the total fare in dollars with Robert's taxi and $y_2$ be that in case of Ben's taxi. Then $y_1$ = $1.75 + 0.65x$ and $y_2$ = $2.50 + 0.50x$. The given condition is, $y_2 \leq y_2$. In other words,

$2.50 + 0.50x \leq 1.75 + 0.65x$

$-0.15x \leq -0.75$

$-x \leq -5$

Or, $x \geq 5$ (note that multiplying both sides by $-1$ reverses the inequality symbol)

So, for a journey of 5 miles and above, hiring Ben's taxi will be cheaper.  The solution is graphed on a number line and shown as below.

Since the answer also includes $5$ miles, there is a closed circle at $5$ on the number line.
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