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Two Step Inequalities

Solving Two-Step Inequalities

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What are linear inequalities and How do they relate to linear equations?

Thursday, February 16th, 2023

Linear Inequalities vs Linear Equations?

Linear Equations are used to describe the equality between two expressions. For example 2x + 1 = x - 3. It is a way of mathematically saying something is equal to something. Sometimes the relationship can be different or more complicated. Common relationships we use to compare quantities in everyday life include, greater than, less than, not equal, between, etc. We use inequalities to represent these sorts of relationships mathematically. You may already be aware of some of the standard symbols used:

Greater than $\gt$
Less than $\lt$
Not equal $\neq$
Greater than or equal $\ge$
Less than or equal $\le$

If the two expressions that are being compared are linear, then the relationship is known as linear inequalities.

Real-world example of how linear inequalities are used to represent constraints or limitations

Example

Example 1. An uber charges a fixed fee of 50 for a ride, plus a rate of 6 per hour. How many hours can you go without exceeding 125?

Answer:

The variable here is the number of hours traveled. Let’s use x to represent it.

The expression that represents the cost of a ride of

x

hours is:

50 + 6x

Since this expression has to be less than or equal to 125, we can use the following inequality to describe the constraint:

50 + 6x \le 125

Solving Linear Inequalities

The algebraic approach to solving inequalities uses the same fundamentals that are used to solve linear equations. The main steps involved are:

$\text{Assign the variable/s (for contextual questions only)}$
$\text{Build the inequality from the context (for contextual questions only)}$
$\text{Rearrange and Simplify like terms}$
$\text{Isolating the variable}$

There are a few important points that you need to keep in mind when rearranging terms and manipulating inequalities as compared to equations. It can be summarized by the rules below:

$\text{Multiplying or dividing an inequality by a negative number reverses the sign of inequality.}$
$\text{Obtaining the reciprocal can reverse the sign of the inequality}$

We can use a simple example to verify rule #1 above.

Proof of Rule #1

Consider two numbers a and b. The diagram shows the positions of these numbers on the number line.

Two numbers a and b marked on a number line

Since b is on the right side of a in the number line, we can represent it using the following inequality:

a \lt b

Now if we consider the negative values of a and b on a number line, their positions will be at a distance of a and b respectively on the left of 0 on the number line.

A number line with the postiive and negative values of a and b marked

So visually it is clear, since -b is on the left of -a on the number line, the inequality that describes the relationship of their sizes would be:

-a \gt -b

If you compare the two inequalities that compare a and b, and -a and -b, it is evident that if you multiply (or divide) by a negative number the sign of the inequality flips.

Before we prove rule #2, let’s try an example of solving two step inequalities.

Example

Example 2. Solve

-12x -18 + 7x \le -3

Answer:

Step 1: Simplify like terms On the left-hand side of the inequality, we have two like terms of x.

-12x + 7x - 18 \le -3

-5x - 18 \le -3

Step 2: Isolate the variable The variable here is x. To isolate the x term we have to first remove -18 from the left-hand side of the inequality by adding 18 to both sides of the inequality.

-5x - 18 + 18 \le -3 + 18

-5x \le 15

Next, we have to remove -5 by dividing both sides by -5.

x \ge -3

Note that the sign of the inequality flipped because we divided by a negative number. (Rule #1)

The proof for rule #2

Let’s consider two positive numbers a and b, where b is greater than a. The inequality that depicts that is :

b \gt a

Next let’s consider c and d, which are the reciprocals of b and a respectively. In a fraction, as the denominator gets larger the size of the fraction reduces. We can use this concept to prove that c will be smaller than d. Now that evaluates to the inequality:

c \lt d

, Which in turn means that,

\frac{1}{b} \lt \frac{1}{a}

This is proof of rule #2. This rule only applies if a and b are both positive or both negative. It’s time to apply the newly gained knowledge to problem-solving.

Example

Example 3.

In a triathlon event, Sarah swims 1.5 km, cycles 40 km, and runs 15 km. Her average cycling speed is twice her average swimming speed, and her average running speed is 1.5 times her average swimming speed. If she finishes the race in less than 3

\frac{1}{3}

hours, what can we say about her average speeds?

Solution:

Step 1. Assign the variables.

Here the variables are Sarah's running speed, cycling speed, and swimming speed. If her swimming speed is s then:

\text{Swimming Speed -- } s

\text{Cycling Speed -- } 2s

\text{Running Speed -- } 1.5s

Step 2. Build the inequality from the context.

The word relationship we have is that:

\text{Swim time} + \text{Run time} + \text{Cycle time} \lt 3\frac{1}{3} \text{hours}

Using the formula :

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

we can represent the relationship mathematically as follows:

\frac{1.5}{s} + \frac{15}{1.5s} + \frac{40}{2s} \lt \frac{10}{3}

Step 3. Simplify like terms By simplifying the fractions we get:

\frac{1.5}{s} + \frac{10}{s} + \frac{20}{s} \lt \frac{10}{3}

\frac{1.5}{s} + \frac{10}{s} + \frac{20}{s} \lt \frac{10}{3}

\frac{31.5}{s} \lt \frac{10}{3}

Step 4. Isolate the variable

The variable here is s. To isolate the term we could obtain the reciprocal on both sides in order to get s on the numerator.

\frac{s}{31.5} \gt \frac{3}{10}

Note that the sign of the inequality flipped because of Rule #2, and it was applicable because we know for certain that the speed "s" is always positive.

Finally, multiply both sides by 31.5

s \gt \frac{3 \times 31.5}{10}

s \gt 9.45

Hence, we can say that

$\text{Sarah's average swimming speed is above 9.45 km/h.}$
$\text{Average cycling speed is above 18.90 km/h.}$
$\text{Sarah's average running speed is greater than 14.18 km/h.}$

Understanding the Solution Set: Inequality Arrows and Interval Notation

Inequality arrows are a method of representing solutions to an inequality visually. There are several conventions that are followed when using arrows:

Interval notation is a common way to represent the solution set of a linear inequality or a system of linear inequalities. It provides a compact and convenient way of writing down the set of all values that satisfy the inequality(ies).

Solution	Interval Notation
$x \gt a$	$({a,\infty})$
$x \lt a$	$(-\infty, a)$
$x \ge a$	$[a, \infty)$
$b \lt x \le a$	$(b, a]$

The square bracket is used to indicate a closed-interval while the parenthesis is used for an open-interval.

Question

Quiz on two step inequalities

Solve the inequality:

2x - 5x \le 5 - 26

and present the solution using the interval notation and inequality arrows.

Solution:

Step 1. Simplify like terms

2x - 5x \le 5 - 26

-3x \le -21

Step 2. Isolate the variable

The variable here is x. To isolate the term we have to divide both sides of the inequality by -3. Remember that according to rule#1 the sign of the inequality reverses.

-3x \le -21

x \ge 7

Representing this using inequality arrows:

In the interval notation, the solution x >=7 would be [7, inf]

Solution	Interval Notation	Inequality arrows
$x \ge 7$	$[{7,\infty})$