Inequalities (OCR GCSE Maths): Revision Notes

Examples:

1. $x<5$ means $x$ can be any number less than $5$, like $4, 0, -23$, etc., but not $5$.
2. $p≥100$ means $p$ can be $100, 1000, 2015$, etc., but not any number less than $100$.
3. $m>−2$ means $m$ is any number greater than $-2$, like $1.9, 0, 4.3,$ but not $-2$!

Examples: 4. $x≥−2$:

• Draw a line starting at $-2$ and going to the right (indicating values greater than $-2$).
• Since the inequality is "greater than or equal to," the circle at $-2$ is filled in.

5. $x<3$:

• Draw a line starting just left of $3$ and extending to the left.
• Because it's "less than" and not "equal to," leave the circle at $3$ unfilled.

6. Combined Inequalities:

• $x>−4$ and $x≤0$:
• Draw a line starting just right of $-4$ to $0$.
• Fill the circle at $0$ (since $x$ can equal $0$) but leave the circle at $-4$ unfilled (since $x$ cannot equal $-4$).

Example: If you start with $−2x≤8$, divide by $-2$ to get $x≥−4$. Notice how the sign flips from $≤$ to $≥$.

Example 1: Solving a Linear Inequality Let's solve the inequality $6x+3≥27$.

Step-by-Step Solution:

1. Isolate the variable term:

• Subtract $3$ from both sides to get $6x≥24$.

2. Solve for the variable:

• Divide both sides by $6$.

Since we divided by a positive number, the inequality sign does not change.

Final Answer: $x≥4$

This means any value of $x$ that is $4$ or greater will satisfy the inequality.

Example 2: Solving an Inequality with Multiple Terms Let's solve the inequality $5x−6<2x+9$.

Step-by-Step Solution:

2. Move all $x$ terms to one side:

• Subtract $2x$ from both sides:

3. Isolate the $x$ term:

• Add $6$ to both sides:

• Then, divide by $3$:

Final Answer: $x<5$

This tells us that $x$ must be less than $5$ to satisfy the inequality.

Example 3: Solving an Inequality Involving a Negative Multiplier Now, let's solve the inequality $−2(5x−4)>98$.

Step-by-Step Solution:

4. Expand the brackets:

• Distribute the $-2$ across the terms inside the brackets:

5. Isolate the $x$ term:

• Subtract $8$ from both sides:

6. Solve for $x$:

• Divide by $-10$, remembering to flip the inequality sign:

Final Answer: $x<−9$

This result tells us that $x$ must be less than $-9$ to satisfy the inequality.

Example 4: Solving a Compound Inequality Let's solve the compound inequality $−4<3x+5≤8$.

Step-by-Step Solution:

7. Unwrap the unknown variable $x$ in the middle:

• Start by subtracting $5$ from all three parts of the inequality:

8. Isolate $x$:

• Now, divide all parts by $3$:

9. Interpret the result:

• This means $x$ must be greater than $-3$ and less than or equal to $1$. Final Answer: $x>−3$ and $x≤1$

Example 1: Solve $x≤2$ 10. Draw the line $x=2$. 11. Notice it's a solid line because the inequality is $≤$. 12. Choose a point on one side of the line:

• Choose ($3,0$), substitute into $x≤2$:

• This point does not satisfy the inequality, so the solution region is on the other side of the line.

Example 2: Solve $y>2x−1$

1. Draw the line $y=2x−1$.
2. Notice it's a dashed line because the inequality is $>$.
3. Choose a point on one side of the line:

• Choose ($2,2$), substitute into $y>2x−1$:

• This point does not satisfy the inequality, so the solution region is on the other side of the line.

Example 3: Solving Multiple Inequalities Together

Given the inequalities $x≥1$, $y≥2$, and $5x+8y≤40$:

4. Solve each inequality graphically:

• For $x≥1$, draw a solid line for $x=1$ and shade to the right.

  

• For $y≥2$, draw a solid line for $y=2$ and shade above.

  

• For $5x+8y≤40$, draw the line $5x+8y=40$ and shade below.

  

5. The solution is the region where all shaded areas overlap.

Example: Solve $2x^2+3≥53$ 7. Simplify the inequality:

Subtract $3$ from both sides:

Divide by $2$:

8. Interpret the inequality:

• We need to find the values of $x$ where $x^2$ is greater than or equal to $25$

9. Sketch the graph:

• Draw the graph of $y=x^2$ and the horizontal line $y=25$.
• The graph shows that $x^2=25$ when $x=5$ or $x=−5$.
• The inequality $x^2≥25$ holds for values where $x$ is either greater than or equal to $5$ or less than or equal to $-5$.

10. Solution:

• From the graph, the solution to the inequality is:

• These are the values of $x$ for which $x^2$ is greater than or equal to $25$.