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Grade 7 Mathematics Learner’s Book

INNODEMS

Section 2.3 Linear Inequalities

Introduction to Linear Inequalities
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In everyday life, we often come across situations where things are not exactly equal but instead follow a comparison. For example, if a bus can carry at most 50 passengers, it means the number of passengers should be less than or equal to 50. Similarly, if a student must be older than 12 years to join a club, it means their age must be greater than 12.
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Mathematics helps us express such comparisons using inequalities.
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An inequality is a mathematical statement that shows the relationship between two values when they are not equal. Unlike equations, which use the equal sign ( = ), inequalities use special symbols to compare values:
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  • Greater than \(( \gt )\text{:}\) Example: \(x \gt 5\) means \(x\) is any number greater than \(5\text{,}\) but \(5\) is not included.
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  • Less than \(( \lt )\text{:}\) Example: \(y \lt 10\) means \(y\) is any number less than \(10\text{,}\) also in this case \(10\) is not included.
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  • Greater than or equal to \(( \ge )\text{:}\) Example: \(p \ge 3p\) means \(p\) is either \(3\) or any number greater than \(3\text{.}\)
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  • Less than or equal to \(( \leq )\text{:}\) Example: \(q \le 8\) means \(q\) is either \(8\) or any number less than \(8\text{.}\)
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What Are Linear Inequalities?
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A linear inequality is an inequality that involves a linear expression. It is similar to a linear equation but uses inequality symbols instead of an equal sign.
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For example:
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  • \(\displaystyle 2x + 3 \ge 7\)
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  • \(\displaystyle 5y - 4 \le 6\)
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Just like linear equations, linear inequalities can be solved by performing operations such as addition, subtraction, multiplication, and division. However, one key difference is that when multiplying or dividing by a negative number, the inequality sign must be reversed.
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Why Are Inequalities Important?
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Inequalities help us solve real-world problems, such as:
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  • Determining budget constraints (e.g., "You can spend at most $100.")
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  • Setting age limits (e.g., "You must be at least 18 years old to vote.")
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  • Calculating time constraints (e.g., "The train arrives in less than 30 minutes.")
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  • Describing physical limits (e.g., "A container can hold no more than 5 liters of water.")
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We are goin to learn how to solve and illustrate linear inequalities in numberlines, understand their applications, and use them to make informed decisions in various situations.
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Subsection 2.3.1 Applying Linear Inequality Symbols to Inequality Statements.

Activity 2.3.1.

Work in Groups.
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  1. You will need a seesaw.
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  2. One learner to sit on one end of the seesaw and another on the other end.
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  3. Find out who is heavier.
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  4. Find out who is lighter.
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  5. Repeat the information using the inequality symbols.
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  6. Share your work with other learners in the class.
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Mary and John on the seesaw.
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Activity 2.3.2.

Work in Groups.
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Read the story below and answer the questions that follow.
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Harriet visited a nutritionist. She was advised not to take more than two eggs in a day. She was also advised to be eating not less than three fruits in a day.
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  1. Why is it important to include fruits in our diet?
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  2. How can we represent the above information using inequality symbols?
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  3. Share your findings with other learners.
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\begin{equation*} \text{Key Takeaway.} \end{equation*}
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\begin{align*} \text{The inequality sign for greater than or equal to is} \ge. \amp \end{align*}
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\begin{align*} \text{Inequality sign for less than or equal to is} (\le). \amp \end{align*}
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Example 2.3.1.

Use the symbols \(\gt\) or \(\lt\) to compare the following pairs of numbers;
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  1. \(\displaystyle 4 \text{ and } 9\)
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  2. \(\displaystyle 16 \text{ and } 11\)
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Solution.
  1. \(4\) is less than \(9\text{.}\)
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    The numbers can be expresed in an algebraic form as;
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    \begin{equation*} 4 < 9 \end{equation*}
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    Also, the numbers can be written as;
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    \begin{equation*} 9 \gt 4 \end{equation*}
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  2. \(16\) is greater than \(11\text{.}\)
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    Therefore, the numbers can be written as;
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    \begin{equation*} 16 > 11 \end{equation*}
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    Also, the numbers can be written as;
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    \begin{equation*} 11 \lt 16 \end{equation*}
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Example 2.3.2.

In a village each family has atleast seven goats. Using letter \(g\) to represent goats, write an inequality to represent the statement given.
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Solution.
Having \(g\) as the number of goats in each family
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Each family should have either \(7\) goats or more.
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Therefore,
\begin{equation*} g = 7 \end{equation*}
\begin{equation*} \text{ or } \end{equation*}
\begin{equation*} g \gt 7 \end{equation*}
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The number of goats (\(g\)) can be represented as an algebraic form as follows;
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\begin{equation*} g > 7 \end{equation*}
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Checkpoint 2.3.3.

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Subsection 2.3.2 Forming a Simple Linear Inequalities in Real Life Situatuans

Activity 2.3.3.

Work in Groups
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Bag \(A\) contains ,\(x\) mangoes. Two mangoes are added to it. The total number of mangoes is ether equal to or less than the number of mangoes in bag \(B\text{,}\) which contains \(30\) mangoes.
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In pairs:
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  1. Add 2 mangoes to \(x\text{.}\)
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  2. Compare the quantities \(x + 2\) and \(30\text{.}\)
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  3. Discuss and share your results.
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Example 2.3.4.

Jane had some oranges. She gave Mwara 5 of them. She was left with at most \(12\) of them. Write an inequality to represent this statement.
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Solution.
Let the number of oranges be \(x\text{.}\)
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Remaining oranges after sharing \(5\) to Mwara will be \(x - 5.\)
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Total oranges Jane had was atleast \(12.\)
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Therefore,
\begin{align*} x - 5 = \amp 12 \end{align*}
or
\begin{align*} x - 5 \lt \amp 12. \end{align*}
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The equations above can be combined so that it can be written in an algebaric form as;
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\begin{align*} x - 5 \le \amp 12. \end{align*}
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Checkpoint 2.3.5.

Exercise here.
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Subsection 2.3.3 Illustrating simple linear inequality on a number line

Number Line.
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A number line is a straight horizontal line used to represent numbers. It helps us visualize numbers, compare values, and understand inequalities easily.
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  • A number line extends infinitely in both directions.
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  • It has a zero \((0)\) in the center, with positive numbers to the right and negative numbers to the left.
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  • The numbers increase as we move to the right and decrease as we move to the left.
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  • The numbers are evenly spaced on the line.
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Take a look at the numberline below:
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Number Line Illustrastion.

Activity 2.3.4.

Work in Groups.
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  1. Consider x > 2.
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  2. Draw a number line.
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  3. Identify some of the numbers that satisfy the inequality.
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  4. Confirm whether the numbers are to the left or right of digit 2.
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  5. Circle digit 2.
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  6. From the circle, draw an arrow facing the direction with numbers which satisfy the inequality.
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  7. Discuss and share with others.
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\begin{equation*} \text{Key Takeaway.} \end{equation*}
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\(\text{For inequalities with symbols} (\lt) \text{or} (\gt), \text{the boundary digits are circled in the representations.}\)
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\(\text{For inequalities with symbols (\le) \text{or} (\ge), the circles are shaded.}\)
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\(\text{We can illustrate inequalities on a number line by understanding the inequality symbols.}\)
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\(\text{An "open point" is used to represent a number that is not included in the inequality.}\)
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\(\text{Take a look at the illustration below;}\)
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\(\text{Illustrate on a number line.} (x > 6)\)
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Open point numberline
\(\text{An "open point" has been used since} 6 \text{in not part of the solution of the inequality.}\)
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\(\text{A "closed point" is used to represent a number that is included in the inequality.}\)
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\(\text{Take a look at the illustration below;}\)
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\begin{equation*} (x \ge 4) \end{equation*}
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closed point inequality
\(\text{A closed point has been used since} (4) \text{is part of the solution of the inequality.}\)
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Example 2.3.6.

Illustrate \(x > 2\) on a number line.
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Solution.
Circle at digit \(2\) and draw a line pointing towards numbers greater than \(2\text{.}\)
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Open point numberline
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Example 2.3.7.

Illustrate \(x \ge 8\) on a number line
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Solution.
Draw a circle at digit \(8\) and shade it then draw a line towards digits greater than \(8\text{.}\)
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closed point inequality
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Checkpoint 2.3.8.

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Subsection 2.3.4 Forming Compound Linear Inequality in one unkown.

Activity 2.3.5.

Work in Groups.
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  1. Consider \(x > 1\) and \(x < 6.\)
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  2. Make \(6\) paper cut-outs and write on each cut-out numbers \(1\) to \(6\text{.}\)
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  3. Arrange the numbers on the cut-outs in an ascending order.
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  4. Shade the cutouts that satisfy \(x > 1\text{.}\)
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  5. Using the same cut-outs, shade the cut-outs that satisfy \(x < 6\text{.}\)
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  6. List down the numbers that satisfy \(x > 1\) and x < 6.
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\begin{equation*} \text{Key Takeaway.} \end{equation*}
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\(\text{When two simple inequalities are combined, they form a compound inequality.}\)
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Example 2.3.9.

Form a compound inequality for \(x \lt 8 \) and \(x \gt 2\text{.}\)
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Solution.
\(x \gt 2\) is equivalent to \(2 \lt x\text{.}\)
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By combining \(2 \lt x\) and \(x \lt 8\text{,}\) we obtain the compound inequality.
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\(2 \lt x \lt 8\text{.}\)
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Example 2.3.10.

Form a compound inequality for \(y \ge 3\) and \(y \le 12.\)
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Solution.
\(y \ge 3\) can be written as \(3 \le y\text{.}\)
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By combining \(3 \le y\) and \(y \le 12\text{,}\) we obtain the compound inequality.
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Therefore, the compound inequality is
\begin{equation*} 3 \le y \le 12\text{.} \end{equation*}
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Checkpoint 2.3.11.

Exercise here.
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Subsection 2.3.5 Illustrating compound inequalities on a number line

Activity 2.3.6. Work in Groups.

  1. Consider
    \begin{equation*} 5<x<12. \end{equation*}
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  2. Write down the two simple inequalities from the compound inequality.
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  3. Draw a number line.
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  4. Illustrate the two inequalities on the same number line.
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  5. Discuss and share with other groups.
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\begin{equation*} \text{Key Takeaway.} \end{equation*}
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Part of a number line that satisfies two linear inequalities is the illustration of a given compound inequality.
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Example 2.3.12.

Illustrate the compound inequality \(4 < y < 8\) on a number line.
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Solution.
The compound inequality \(4 < y < 8\) can be split into two simple inequalities: \(y > 4\) and \(y < 8\text{.}\) To illustrate this on a number line, draw open circles at \(4\) and \(8\text{.}\)
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numberline with shaded region in between
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Checkpoint 2.3.13.

Exercise here.
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Activity 2.3.7. Work in Groups.

  1. Consider \(3 \le x \le 9\text{.}\)
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  2. Write down the two simple inequalities from the compound inequality.
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  3. Draw a number line.
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  4. Illustrate the two inequalities on the same number line.
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  5. Discuss and share with other groups.
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\begin{equation*} \text{Key Takeaway.} \end{equation*}
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To represent a compound inequality on a number line, illustrate the regions that satisfy each inequality and highlight the overlapping section.
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Example 2.3.14.

Illustrate the compound inequality \(8 \le x \le 12\) on a number line.
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Solution.
The compound inequality \(8 \le x \le 12\) can be split into two simple inequalities: \(x \ge 8\) and \(x \le 12\text{,}\) draw closed points and then shade the region between them..
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Number line
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Checkpoint 2.3.15.

Exercise here.
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