Solve the Inequality for h 4(3h-7)< =20(h-1)

Solution:

Step 1: Expand the left-hand side expression

• Step 1.1: Start with the original inequality. Add zero terms if necessary for clarity.

  $$0+0+4(3h-7)\le 20(h-1)$$

• Step 1.2: Recognize that adding zero does not change the inequality.

  $$4(3h-7)\le 20(h-1)$$

• Step 1.3: Use the distributive property to expand the terms.

  $$4\cdot 3h+4\cdot (-7)\le 20(h-1)$$

• Step 1.4: Perform the multiplication.

  • Step 1.4.1: Multiply 3 by 4.

    $$12h+4\cdot (-7)\le 20(h-1)$$

  • Step 1.4.2: Multiply 4 by -7.

    $$12h-28\le 20(h-1)$$

Step 2: Expand the right-hand side expression

• Step 2.1: Apply the distributive property.

  $$12h-28\le 20h+20\cdot (-1)$$

• Step 2.2: Multiply 20 by -1.

  $$12h-28\le 20h-20$$

Step 3: Isolate terms with h on one side

• Step 3.1: Subtract 20h from both sides to move h terms to the left.

  $$12h-28-20h\le -20$$

• Step 3.2: Combine like terms.

  $$-8h-28\le -20$$

Step 4: Isolate constant terms on the other side

• Step 4.1: Add 28 to both sides to move constants to the right.

  $$-8h\le -20+28$$

• Step 4.2: Combine the constants.

  $$-8h\le 8$$

Step 5: Solve for h

• Step 5.1: Divide by -8, remembering to reverse the inequality.

  $$\frac{-8h}{-8}\ge \frac{8}{-8}$$

• Step 5.2: Simplify both sides.

  • Step 5.2.1: Cancel out the -8 on the left.

    • Step 5.2.1.1: Cancel the common factors.

      $$\frac{\cancel{-8}h}{\cancel{-8}}\ge \frac{8}{-8}$$

    • Step 5.2.1.2: Simplify to h.

      $$h\ge \frac{8}{-8}$$

  • Step 5.3: Simplify the right side.

    • Step 5.3.1: Divide 8 by -8.

      $$h\ge -1$$

Step 6: Express the solution in different forms

• Inequality Form:

  $$h\ge -1$$

• Interval Notation:

  $$[-1,\mathrm{\infty })$$

Knowledge Notes:

To solve the inequality $4(3h-7)\le 20(h-1)$, we followed a structured approach:

1. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to expand expressions by multiplying each term inside the parentheses by the factor outside.

2. Combining Like Terms: Terms that contain the same variables to the same power can be combined by adding or subtracting their coefficients.

3. Inequality Manipulation: When solving inequalities, you can add, subtract, multiply, or divide both sides by the same number. However, when multiplying or dividing by a negative number, the direction of the inequality sign must be reversed.

4. Interval Notation: This is a way of writing the set of all numbers between two endpoints. The notation $[a,b)$ means the interval includes $a$ and goes up to, but does not include, $b$.

5. Negative Division: When dividing both sides of an inequality by a negative number, the inequality sign flips direction, which is a critical step in ensuring the correct solution.

By following these steps and applying the relevant mathematical rules, we can solve the inequality and express the solution in various forms.