Solving and Graphing Inequalities

To solve and graph inequalities, follow these steps:

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1. Solving Linear Inequalities:

The process of solving inequalities is similar to solving equations, but with a key difference: when you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign.

Example:

Solve the inequality $2x + 3 > 7$:

• Step 1: Subtract 3 from both sides:
  $2x + 3 - 3 > 7 - 3$
  $2x > 4$

• Step 2: Divide both sides by 2:
  $\frac{2x}{2} > \frac{4}{2}$
  $x > 2$

The solution is $x > 2$, meaning $x$ must be greater than 2.

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2. Solving Inequalities with Negative Multiplication/Division:

When multiplying or dividing by a negative number, the inequality sign flips.

Example:

Solve $-3x \leq 9$:

• Step 1: Divide both sides by -3 (and flip the inequality):
  $\frac{-3x}{-3} \geq \frac{9}{-3}$
  $x \geq -3$

The solution is $x \geq -3$.

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3. Graphing Linear Inequalities on a Number Line:

• If the inequality involves $<$ or $>$, use an open circle on the boundary point (indicating that the boundary point is not included).
• If the inequality involves $\leq$ or $\geq$, use a closed circle on the boundary point (indicating that the boundary point is included).
• Then, shade the region of the number line where the inequality is satisfied.

For $x > 2$, you would:

• Draw an open circle at $x = 2$ (since $2$ is not included).
• Shade everything to the right of $x = 2$.

For $x \geq -3$, you would:

• Draw a closed circle at $x = -3$ (since $-3$ is included).
• Shade everything to the right of $x = -3$.

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4. Solving and Graphing Compound Inequalities:

Compound inequalities involve two inequalities joined by "and" or "or".

• For "and" inequalities (e.g., $a < x \leq b$), the solution is the overlap (intersection) of the solutions of both inequalities.
• For "or" inequalities (e.g., $x < a$ or $x > b$), the solution is the union of the solutions of both inequalities.

Example:

Solve and graph $1 \leq x < 5$:

• Step 1: Solve each inequality:
  $1 \leq x$ and $x < 5$

• Step 2: Represent the solution on a number line:
  Use a closed circle at $x = 1$ and an open circle at $x = 5$, then shade the region between $1$ and $5$.

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Example Graphs:

1. $x > 2$: Open circle at 2, shade to the right.
2. $x \leq -1$: Closed circle at -1, shade to the left.
3. Compound: $-2 < x \leq 3$: Open circle at -2, closed circle at 3, shade in between.

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Any questions so far or would you like further details?

Related Questions:

1. How do you solve inequalities involving fractions or rational expressions?
2. What is the difference between "and" and "or" in compound inequalities?
3. How can you graph inequalities with two variables?
4. Can inequalities have no solution? If so, how do you identify them?
5. How do you solve absolute value inequalities?

Tip:

Remember to always reverse the inequality sign when multiplying or dividing by a negative number!