Introduction
- Inequalities are similar to equations. While equations have specific values that satisfy the equation, inequalities have a feasible region (a set of values satisfying the inequality).
- Example of Inequality:
- The solution to these inequalities exists in the form of intervals (a set of numbers that lies between two values).
Example: $(-12, 1]$. The number is greater than $-12$ and less than or equal to $1$.
Watch: Linear Inequalities
Linear Inequalities
Linear inequalities are inequalities in which the maximum power raised to a variable is one.
Steps to Solve Linear Inequality:
Find the value of the variable using addition, subtraction, multiplication, or division on both sides of the inequality so that the variable becomes isolated.
For the expression $2x + 9 < 15$:
(Whenever we change the sign on both sides, the symbol is reversed from greater than to less than and vice versa).
Express the solution in the form of an interval or on a number line: $x = (-\infty, 3)$
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Write down the inequality shown on the number line.
Identify the variable: Notice that the number line is labelled with the variable $y$ on the far right.
Examine the lower bound:
- There is a solid (filled) circle at $-3$.
- A solid circle indicates that the number is included in the range, meaning we use the “less than or equal to” symbol ($\le$).
Examine the upper bound:
- There is also a solid (filled) circle at $4$.
- This indicates that $4$ is included in the range as well, meaning we again use the “less than or equal to” symbol ($\le$).
Combine into a compound inequality:
The line connects $-3$ and $4$, meaning $y$ represents all values between and including $-3$ and $4$. Therefore, the inequality is:
Final Answer: $-3 \le y \le 4$
Write down the inequality shown on the number line.
Identify the variable: Notice that the number line is labelled with the variable $y$ on the right side.
Examine the lower bound:
- There is a solid (filled) circle at $-4$.
- This indicates that $-4$ is included in the range, meaning we use the “less than or equal to” symbol ($\le$).
Examine the upper bound:
- There is an open (unfilled) circle at $3$.
- This indicates that $3$ is not included in the range, meaning we use the strictly “less than” symbol ($<$).
Combine into a compound inequality:
The line connects $-4$ and $3$, meaning $y$ takes all values between them (including $-4$, but excluding $3$). Therefore, the inequality is:
Final Answer: $-4 \le y < 3$
Write down the inequality shown on the number line.
Identify the circle’s position: The circle is positioned above the number $2$ on the number line.
Determine the type of inequality:
- The circle is open (unfilled), which means the inequality does not include the number $2$ itself.
- Therefore, we use strictly less than ($<$) or strictly greater than ($>$).
Determine the direction of the arrow: The arrow points to the left towards smaller numbers (negative infinity).
Write the inequality:
Since the arrow points to all numbers strictly less than $2$, the inequality is:
Final Answer: $x < 2$
(a) On the number line, show the inequality $n < 2$.
(b) $4 \le y < 8$ where $y$ is an integer. Write down all the possible values of $y$.
(c) Solve $4x + 6 \le x + 21$.
(a)
Identify the circle type: The inequality uses the strictly less than symbol ($<$), so place an open (unfilled) circle above $2$ on the number line.
Determine the direction: Since $n$ is less than $2$, draw a line or arrow extending from the circle to the left (towards smaller numbers).
Final Answer (a): Open circle at 2, arrow to the left
(b)
Identify the bounds:
- The lower bound symbol $\le$ means that $4$ is included in the list of values.
- The upper bound symbol $<$ means that $8$ is not included in the list of values.
List the integers: The whole numbers within this range are $4, 5, 6, 7$.
Final Answer (b): $4, 5, 6, 7$
(c)
Subtract $x$ from both sides:
Subtract 6 from both sides:
Divide both sides by 3:
Final Answer (c): $x \le 5$
(a) On the number line, show the inequality $x > -3$.
(b) $1 \le y < 5$ where $y$ is an integer. Write down all the possible values of $y$.
(c) Solve $4t + 7 \le 19$.
(a)
Describe the number line:
- The inequality uses the strictly greater than symbol ($>$), so an open (unfilled) circle is placed above $-3$.
- Since $x$ is greater than $-3$, draw an arrow extending from the circle to the right.
Final Answer (a): Open circle at -3, arrow to the right
(b)
List the integers:
- The lower bound symbol $\le$ means that $1$ is included in the list of values.
- The upper bound symbol $<$ means that $5$ is not included in the list of values.
- The whole numbers within this range are $1, 2, 3, 4$.
Final Answer (b): $1, 2, 3, 4$
(c)
Solve the linear inequality:
Subtract $7$ from both sides:
Divide both sides by $4$:
Final Answer (c): $t \le 3$
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