Skip to main content

Grade 9 Maths Book An Easy Way to Learn Maths

INNODEMS

Section 2.3 Linear Inequalities

Subsection 2.3.1 Interpreting Linear Inequalities in One and Two Unknowns

In daily life, we often work with conditions or limits, but sometimes we work with exact values like "You must be at least \(18\) years old to vote" or "The bag should not weigh more than \(10\) kg".
🔗
A \(\textbf{linear inequality}\) is a mathematical sentence that uses inequality symbols instead of the equals sign. These include:
🔗
Linear inequalities can have one unknown (e.g., \(x \ge 10\)) or two unknowns (e.g., \(y \lt 2x + 5\)), and they represent a range of possible values that satisfy the condition.
🔗

Activity 2.3.1.

Work in groups
🔗
What You need:
🔗
🔗
Instructions:
🔗
Discuss the following real-life situations and write an appropriate linear inequality for each.
🔗
  1. A student must score at least \(50\) marks to pass an exam. Let \(x\) be the marks scored.
    🔗
  2. A matatu cannot carry more than \(14\) passengers. Let \(p\) be the number of passengers.
    🔗
  3. A phone costs more than Ksh \(5,000\text{.}\) Let \(c\) be the cost.
    🔗
  4. The total luggage weight must be less than or equal to \(25\) kg. Let \(w\) be the weight.
    🔗
🔗
After writing the inequalities:
  • Identify whether each inequality has one or two unknowns.
    🔗
  • Explain your interpretation to another group and listen to theirs.
    🔗
🔗
🔗
Key Takeaway:
🔗
Linear inequalities describe limitations or conditions in real-life situations. They allow us to work with ranges of possible values rather than just one answer.
🔗

Example 2.3.1.

A parent gives a student Ksh \(500\) for lunch for the entire week. If lunch costs \(x\) shillings per day and they have \(5\) school going days, write an inequality to represent how much can be spent per day by the student.
🔗
Solution.
The total cost for \(5\) days is \(5x\text{.}\) The student cannot spend more than Ksh \(500\text{,}\) so:
🔗
\(5x \le 500\)
🔗
Solving the inequality:
🔗
\begin{gather*} 5x \le 500\\ x \le 100 \end{gather*}
🔗
This means the student can spend at most Ksh \(100\) each day.
🔗
🔗
🔗

Exercises Exercises

1.
Answer the following questions in your exercise book.
🔗
  1. Write inequalities to represent the following statements:
    🔗
    1. You must be at least \(18\) years old to register to vote. Let \(a\) represent age.
      🔗
      🔗
    2. You want to buy \(x\) pens at Ksh \(25\) each. You have Ksh \(210\text{.}\)
      🔗
      🔗
    3. A shopkeeper must sell more than \(60\) items to make a profit. Let \(n\) be the number of items sold.
      🔗
      🔗
    4. A student may work at most \(40\) hours per week. Let \(h\) be the number of hours worked.
      🔗
      🔗
    🔗
    🔗
  2. A trip costs Ksh \(800\) per student. The total cost should not exceed Ksh \(16,000\text{.}\) Let \(t\) be the number of students. Write and interpret the inequality.
    🔗
    🔗
🔗
🔗
🔗
🔗

Subsection 2.3.2 Solving Linear Inequalities in One and Two Unknowns

Solving a linear inequality is similar to solving an equation, except the solution is often a range of values. These values can be shown on a number line (for one variable) or on a coordinate plane (for two variables). Remember, when multiplying or dividing both sides of an inequality by a negative number, the inequality sign reverses direction.
🔗

Activity 2.3.2.

Work in groups
🔗
What you need:
🔗
🔗
Instructions:
🔗
Consider the following inequalities:
🔗
  1. \(\displaystyle x + 4 < 9\)
    🔗
  2. \(\displaystyle 2x \ge 12\)
    🔗
  3. \(\displaystyle -3x \gt 6\)
    🔗
  4. \(\displaystyle \frac{x - 2}{3} \le 5\)
    🔗
🔗
Solve each inequality and represent the solution on a number line.
🔗
  • For each inequality, discuss with your group how the solution set is different from that of an equation.
    🔗
  • Share your number line diagrams with another group and compare your ranges.
    🔗
🔗
🔗
Key Takeaway:
🔗
Solving linear inequalities involves similar steps to solving equations, but always remember to reverse the inequality when multiplying or dividing by a negative number. For two-variable inequalities, graph the boundary and shade the region of solutions.
🔗

Example 2.3.2.

Solve and represent the solution to the inequality: \(3x - 5 \le 10\) on a numberline.
🔗
Solution.
\begin{gather*} 3x - 5 \le 10\\ 3x \le 15\\ x \le 5 \end{gather*}
🔗
The solution is all values of \(x\) less than or equal to \(5\text{.}\)
🔗
On a number line, this is shown as a solid dot at \(5\) and shading to the left.
🔗
🔗
🔗

Activity 2.3.3.

Work in groups
🔗
What you need:
🔗
🔗
Instructions:
🔗
  1. Plot the boundary line for the inequality \(y \lt 2x + 3\text{.}\) Use a dashed line since the inequality does not include equality.
    🔗
  2. Pick a test point like \((0,0)\) to determine which side of the line to shade.
    🔗
  3. Shade the correct region that represents all solutions to the inequality.
    🔗
  4. Repeat the process for \(y \ge -x + 1\text{,}\) this time using a solid line.
    🔗
  5. Discuss the meaning of overlapping shaded regions if both inequalities were given together.
    🔗
🔗
🔗

Example 2.3.3.

Solve and graph the inequality: \(y \le -3x + 1\text{.}\)
🔗
Solution.
Step 1: Identify the boundary line \(y = -3x + 1\text{.}\) Since the inequality is \(\le\text{,}\) we will draw a solid line.
🔗
Step 2: Choose a test point like \((0,0)\text{:}\)
🔗
\(0 \le -3(0) + 1 → 0 \le 1\text{,}\) which is true.
🔗
Step 3: Shade the region below the line, since it represents all values of \(y\) less than or equal to \(-3x + 1\text{.}\)
🔗
The graph shows the line and the shaded region representing the solution set.
🔗
🔗
🔗

Exercises Exercises

1.
\(\textbf{Work on the following questions in your notebook.}\)
🔗
  1. Solve and represent the solution on a number line: \(2x + 3 \gt 7\)
    🔗
  2. Solve and draw a number line for: \(-4x \le 12\)
    🔗
  3. Plot the solution set for \(y \gt x - 2\) on a graph.
    🔗
  4. Plot the region that satisfies both \(y \le 2x + 1\) and \(y \gt -x\text{.}\)
    🔗
  5. Write a real-life situation that can be modeled using the inequality \(x \le 6\) and explain your reasoning.
    🔗
🔗
🔗
🔗

Subsection 2.3.3 Representing Linear Inequalities Graphically

In many real-life situations, we deal with inequalities such as limits, budgets, or boundaries. Graphing these inequalities helps us visualize all the possible solutions that satisfy the conditions.
🔗
When a linear inequality involves two variables (like \(x\) and \(y\)), its solution is a region in the coordinate plane, not just a line.
🔗
The boundary line helps define where the inequality changes from true to false. We use:
  • Solid lines for inequalities like \(\le\) or \(\ge\)
    🔗
  • Dashed lines for inequalities like \(\lt\) or \(\gt\)
    🔗
🔗

Example 2.3.4.

Graph the inequality \(x \le 1\) on a number line.
🔗
Solution.
Draw a number line with values from \(-5\) to \(5\text{.}\) Put a solid circle at \(1\) and shade everything to the left.
🔗
🔗
🔗

Activity 2.3.5.

Work in groups
🔗
What you need:
🔗
🔗
Instructions:
🔗
  1. Draw the line \(y = -x + 2\text{.}\) Use a dashed line since the inequality will not include equality.
    🔗
  2. Test a point, like \((0,0)\text{.}\) Substitute into \(y \gt -x + 2\text{.}\)
    🔗
  3. Since \(0 \gt -0 + 2\) is false, shade the opposite side from \((0,0)\text{.}\)
    🔗
  4. Repeat the same process for \(y \le 2x - 1\text{.}\)
    🔗
🔗
🔗
Key Takeaway:
🔗
Inequalities in two variables describe regions of the graph. Always draw the boundary line first, then use a test point to decide which side to shade. Use dashed or solid lines to indicate strict or inclusive inequalities.
🔗

Exercises Exercises

1.
\(\textbf{Answer the following in your exercise book.}\)
🔗
  1. Draw a number line and graph the inequality \(x \lt -1\text{.}\)
    🔗
  2. Represent \(y \ge 2x - 3\) on the Cartesian plane.
    🔗
  3. Graph the inequality \(y \lt -\frac{1}{2}x + 4\text{.}\)
    🔗
  4. Write two real-life examples that can be represented using inequalities.
    🔗
  5. If \(x \ge 2\) and \(x \lt 5\text{,}\) shade the region on the number line that satisfies both.
    🔗
🔗
🔗
🔗
🔗
PrevTopNext