Algebraic inequalities practice problems for solving linear equations.

It looks like you've uploaded a list of inequalities (numbered 15 to 26), and you'd like me to solve them and explain the solutions. Since I can't see the image directly, I’ll assume you’re referring to the text you’ve provided — a list of 12 linear inequalities.

Let’s go through each inequality one by one, solve it step by step, and explain the process.

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15) $-2 < \frac{x}{4}$

Step 1: Multiply both sides by 4 to eliminate the denominator:
$$
4 \cdot (-2) < x \Rightarrow -8 < x
$$

Answer: $x > -8$

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16) $-2x + 5 < 9$

Step 1: Subtract 5 from both sides:
$$
-2x < 4
$$

Step 2: Divide both sides by -2 (remember: when dividing by a negative number, reverse the inequality):
$$
x > -2
$$

Answer: $x > -2$

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17) $-12 < 12 - 4x$

Step 1: Subtract 12 from both sides:
$$
-24 < -4x
$$

Step 2: Divide both sides by -4 (reverse inequality):
$$
6 > x \quad \text{or} \quad x < 6
$$

Answer: $x < 6$

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18) $2 - \frac{4}{5}x < 6$

Step 1: Subtract 2 from both sides:
$$
-\frac{4}{5}x < 4
$$

Step 2: Multiply both sides by 5 to eliminate denominator:
$$
-4x < 20
$$

Step 3: Divide by -4 (reverse inequality):
$$
x > -5
$$

Answer: $x > -5$

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19) $\frac{1}{3}x = \frac{1}{3} - \frac{2x}{3}$

Wait! This is not an inequality, it's an equation. But it's listed with inequalities. Let's solve it as an equation:

Multiply both sides by 3 to eliminate denominators:
$$
x = 1 - 2x
$$

Add $2x$ to both sides:
$$
3x = 1 \Rightarrow x = \frac{1}{3}
$$

But since this is not an inequality, maybe there was a typo. If it were an inequality (e.g., $<$ or $>$), we’d solve similarly. As written, it's an equation.

Answer: $x = \frac{1}{3}$

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20) $-\frac{1}{2}x - 2 > 3$

Step 1: Add 2 to both sides:
$$
-\frac{1}{2}x > 5
$$

Step 2: Multiply both sides by -2 (reverse inequality):
$$
x < -10
$$

Answer: $x < -10$

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21) $2(4x - 1) < 38$

Step 1: Distribute:
$$
8x - 2 < 38
$$

Step 2: Add 2 to both sides:
$$
8x < 40
$$

Step 3: Divide by 8:
$$
x < 5
$$

Answer: $x < 5$

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22) $-9 - \frac{x}{2} \geq x + 1$

Step 1: Eliminate fractions by multiplying all terms by 2:
$$
-18 - x \geq 2x + 2
$$

Step 2: Move variables to one side:
$$
-18 - 2 \geq 2x + x \Rightarrow -20 \geq 3x
$$

Step 3: Divide by 3:
$$
x \leq -\frac{20}{3}
$$

Answer: $x \leq -\frac{20}{3}$

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23) $1 - \frac{3}{2}x \geq x - 4$

Step 1: Multiply all terms by 2 to eliminate fractions:
$$
2 - 3x \geq 2x - 8
$$

Step 2: Move variables to one side:
$$
2 + 8 \geq 2x + 3x \Rightarrow 10 \geq 5x
$$

Step 3: Divide by 5:
$$
x \leq 2
$$

Answer: $x \leq 2$

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24) $4(x + 1) < 2x + 3$

Step 1: Expand:
$$
4x + 4 < 2x + 3
$$

Step 2: Subtract $2x$ from both sides:
$$
2x + 4 < 3
$$

Step 3: Subtract 4:
$$
2x < -1
$$

Step 4: Divide by 2:
$$
x < -\frac{1}{2}
$$

Answer: $x < -\frac{1}{2}$

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25) $-(x - 1) - 4x > x - 3$

Step 1: Simplify left side:
$$
- x + 1 - 4x > x - 3 \Rightarrow -5x + 1 > x - 3
$$

Step 2: Subtract $x$ from both sides:
$$
-6x + 1 > -3
$$

Step 3: Subtract 1:
$$
-6x > -4
$$

Step 4: Divide by -6 (reverse inequality):
$$
x < \frac{2}{3}
$$

Answer: $x < \frac{2}{3}$

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26) $3 - 8(x + 1) \leq -13x$

Step 1: Expand:
$$
3 - 8x - 8 \leq -13x \Rightarrow -8x - 5 \leq -13x
$$

Step 2: Add $13x$ to both sides:
$$
5x - 5 \leq 0
$$

Step 3: Add 5:
$$
5x \leq 5
$$

Step 4: Divide by 5:
$$
x \leq 1
$$

Answer: $x \leq 1$

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✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| 15) | $x > -8$ |
| 16) | $x > -2$ |
| 17) | $x < 6$ |
| 18) | $x > -5$ |
| 19) | $x = \frac{1}{3}$ (Equation, not inequality) |
| 20) | $x < -10$ |
| 21) | $x < 5$ |
| 22) | $x \leq -\frac{20}{3}$ |
| 23) | $x \leq 2$ |
| 24) | $x < -\frac{1}{2}$ |
| 25) | $x < \frac{2}{3}$ |
| 26) | $x \leq 1$ |

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Let me know if you'd like these graphed on a number line or expressed in interval notation!