Spencer Reese
08/11/2023 · Primary School
(a) \( 7(x+4)+5 \leq(x-7) \) or \( 8(x+4)+5>-4 x+4 \) First solve the individual inequalities. Apply the distribu \[ 7 x+\square \leq x-7 \text { or } 8 x+\square>-4 x+4 \]
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Tutor-Verified Answer
Step-by-step Solution
Let's solve the individual inequalities step by step.
### First Inequality:
\[ 7(x + 4) + 5 \leq (x - 7) \]
1. Distribute \(7\) on the left side:
\[
7x + 28 + 5 \leq x - 7
\]
This simplifies to:
\[
7x + 33 \leq x - 7
\]
2. Now, subtract \(x\) from both sides:
\[
7x - x + 33 \leq -7
\]
Which simplifies to:
\[
6x + 33 \leq -7
\]
3. Next, subtract \(33\) from both sides:
\[
6x \leq -7 - 33
\]
This simplifies to:
\[
6x \leq -40
\]
4. Finally, divide both sides by \(6\):
\[
x \leq -\frac{40}{6}
\]
Simplifying gives:
\[
x \leq -\frac{20}{3}
\]
### Second Inequality:
\[ 8(x + 4) + 5 > -4x + 4 \]
1. Distribute \(8\) on the left side:
\[
8x + 32 + 5 > -4x + 4
\]
This simplifies to:
\[
8x + 37 > -4x + 4
\]
2. Now, add \(4x\) to both sides:
\[
8x + 4x + 37 > 4
\]
Which simplifies to:
\[
12x + 37 > 4
\]
3. Next, subtract \(37\) from both sides:
\[
12x > 4 - 37
\]
This simplifies to:
\[
12x > -33
\]
4. Finally, divide both sides by \(12\):
\[
x > -\frac{33}{12}
\]
Simplifying gives:
\[
x > -\frac{11}{4}
\]
### Summary of Solutions:
- From the first inequality, we have:
\[
x \leq -\frac{20}{3}
\]
- From the second inequality, we have:
\[
x > -\frac{11}{4}
\]
Thus, the solution to the compound inequality is:
\[
x \leq -\frac{20}{3} \quad \text{or} \quad x > -\frac{11}{4}
\]
Quick Answer
The solution to the compound inequality is \( x \leq -\frac{20}{3} \) or \( x > -\frac{11}{4} \).
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