Answer :
Sure! Let's solve each inequality step-by-step to determine the correct answers.
### 1. Solve [tex]\( -2(2x + 5) \leq -7(x + 4) \)[/tex]
First, expand both sides:
[tex]\[ -2(2x + 5) = -4x - 10 \][/tex]
[tex]\[ -7(x + 4) = -7x - 28 \][/tex]
So the inequality becomes:
[tex]\[ -4x - 10 \leq -7x - 28 \][/tex]
Next, add [tex]\( 7x \)[/tex] to both sides:
[tex]\[ 3x - 10 \leq -28 \][/tex]
Then, add 10 to both sides:
[tex]\[ 3x \leq -18 \][/tex]
Finally, divide both sides by 3:
[tex]\[ x \leq -6 \][/tex]
Thus, [tex]\[ x \leq -6 \][/tex].
### 2. Solve [tex]\( \frac{4x - 6}{2} \geq 3x - 1 \)[/tex]
Simplify the fraction:
[tex]\[ 2x - 3 \geq 3x - 1 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ - x - 3 \geq -1 \][/tex]
Add 3 to both sides:
[tex]\[ - x \geq 2 \][/tex]
Multiply both sides by -1, and reverse the inequality sign:
[tex]\[ x \leq -2 \][/tex]
Thus, [tex]\[ x \leq -2 \][/tex].
### 3. Solve [tex]\( 5(x - 3) \leq 9(x + 1) \)[/tex]
First, expand both sides:
[tex]\[ 5(x - 3) = 5x - 15 \][/tex]
[tex]\[ 9(x + 1) = 9x + 9 \][/tex]
So the inequality becomes:
[tex]\[ 5x - 15 \leq 9x + 9 \][/tex]
Next, subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ -15 \leq 4x + 9 \][/tex]
Then, subtract 9 from both sides:
[tex]\[ -24 \leq 4x \][/tex]
Finally, divide both sides by 4:
[tex]\[ -6 \leq x \][/tex]
or equivalently
[tex]\[ x \geq -6 \][/tex]
Thus, [tex]\[ x \geq -6 \][/tex].
### 4. Solve [tex]\( 3(x - 4) + 5 \geq 2x - 9 \)[/tex]
Expand:
[tex]\[ 3(x - 4) + 5 = 3x - 12 + 5 = 3x - 7 \][/tex]
So the inequality becomes:
[tex]\[ 3x - 7 \geq 2x - 9 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ x - 7 \geq -9 \][/tex]
Add 7 to both sides:
[tex]\[ x \geq -2 \][/tex]
Thus, [tex]\[ x \geq -2 \][/tex].
### Matching the inequalities to the solutions:
[tex]\[ \begin{array}{ll} -2(2 x+5) \leq-7(x+4) & x \leq-6 \\ \frac{4 x-6}{2} \geq 3 x-1 & x \leq-2 \\ 5(x-3) \leq 9(x+1) & x \geq-6 \\ 3(x-4)+5 \geq 2 x-9 & x \geq-2 \\ \end{array} \][/tex]
Thus, the correct matches are:
[tex]\[ \begin{array}{ll} -2(2 x+5) \leq-7(x+4) & x \leq-6 \\ \frac{4 x-6}{2} \geq 3 x-1 & x \leq-2 \\ 5(x-3) \leq 9(x+1) & x \geq-6 \\ 3(x-4)+5 \geq 2 x-9 & x \geq-2 \\ \end{array} \][/tex]
### 1. Solve [tex]\( -2(2x + 5) \leq -7(x + 4) \)[/tex]
First, expand both sides:
[tex]\[ -2(2x + 5) = -4x - 10 \][/tex]
[tex]\[ -7(x + 4) = -7x - 28 \][/tex]
So the inequality becomes:
[tex]\[ -4x - 10 \leq -7x - 28 \][/tex]
Next, add [tex]\( 7x \)[/tex] to both sides:
[tex]\[ 3x - 10 \leq -28 \][/tex]
Then, add 10 to both sides:
[tex]\[ 3x \leq -18 \][/tex]
Finally, divide both sides by 3:
[tex]\[ x \leq -6 \][/tex]
Thus, [tex]\[ x \leq -6 \][/tex].
### 2. Solve [tex]\( \frac{4x - 6}{2} \geq 3x - 1 \)[/tex]
Simplify the fraction:
[tex]\[ 2x - 3 \geq 3x - 1 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ - x - 3 \geq -1 \][/tex]
Add 3 to both sides:
[tex]\[ - x \geq 2 \][/tex]
Multiply both sides by -1, and reverse the inequality sign:
[tex]\[ x \leq -2 \][/tex]
Thus, [tex]\[ x \leq -2 \][/tex].
### 3. Solve [tex]\( 5(x - 3) \leq 9(x + 1) \)[/tex]
First, expand both sides:
[tex]\[ 5(x - 3) = 5x - 15 \][/tex]
[tex]\[ 9(x + 1) = 9x + 9 \][/tex]
So the inequality becomes:
[tex]\[ 5x - 15 \leq 9x + 9 \][/tex]
Next, subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[ -15 \leq 4x + 9 \][/tex]
Then, subtract 9 from both sides:
[tex]\[ -24 \leq 4x \][/tex]
Finally, divide both sides by 4:
[tex]\[ -6 \leq x \][/tex]
or equivalently
[tex]\[ x \geq -6 \][/tex]
Thus, [tex]\[ x \geq -6 \][/tex].
### 4. Solve [tex]\( 3(x - 4) + 5 \geq 2x - 9 \)[/tex]
Expand:
[tex]\[ 3(x - 4) + 5 = 3x - 12 + 5 = 3x - 7 \][/tex]
So the inequality becomes:
[tex]\[ 3x - 7 \geq 2x - 9 \][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[ x - 7 \geq -9 \][/tex]
Add 7 to both sides:
[tex]\[ x \geq -2 \][/tex]
Thus, [tex]\[ x \geq -2 \][/tex].
### Matching the inequalities to the solutions:
[tex]\[ \begin{array}{ll} -2(2 x+5) \leq-7(x+4) & x \leq-6 \\ \frac{4 x-6}{2} \geq 3 x-1 & x \leq-2 \\ 5(x-3) \leq 9(x+1) & x \geq-6 \\ 3(x-4)+5 \geq 2 x-9 & x \geq-2 \\ \end{array} \][/tex]
Thus, the correct matches are:
[tex]\[ \begin{array}{ll} -2(2 x+5) \leq-7(x+4) & x \leq-6 \\ \frac{4 x-6}{2} \geq 3 x-1 & x \leq-2 \\ 5(x-3) \leq 9(x+1) & x \geq-6 \\ 3(x-4)+5 \geq 2 x-9 & x \geq-2 \\ \end{array} \][/tex]
